____ THE ______ MATHEMATICS ___ _________ EDUCATOR _____ Volume 16 Number 2

Fall 2006

MATHEMATICS EDUCATION STUDENT ASSOCIATION THE UNIVERSITY OF GEORGIA

Editorial Staff

A Note from the Editor

Editor Kyle T. Schultz

Dear TME Reader,

Associate Editors Rachael Brown Na Young Kwon Catherine Ulrich Production Ginger A. Rhodes Advisor Dorothy Y. White

MESA Officers 2005-2006 President Rachael Brown Vice-President Filyet Asli Ersoz Secretary Eileen Murray Treasurer Jun-ichi Yamaguchi NCTM Representative Ginger A. Rhodes Undergraduate Representative Laine Bradshaw Rachel Stokely

Along with the editorial team, I am very happy to present a new issue of The Mathematics Educator, the second and final issue of Volume 16. In it, I hope you will find research and commentary that spark new ideas and build upon the existing dialogue and research of the mathematics education community. In accordance with the mission of TME, this issue presents a variety of viewpoints on a broad spectrum of issues in mathematics education. From his observation of middle school students working on algebraic tasks, David Slavit explores the mechanisms through which these students create meaning from their endeavors and notice algebraic properties. Mehmet Ozturk and Kusum Singh share research on the effects of various socioeconomic factors, such as parental involvement, educational aspirations, and mathematics selfconcept, upon the choices students make with regard to taking advanced high school mathematics courses. This issue also features articles and editorials focusing upon teacher education. Lovemore Nyaumwe and David Mtetwa present their investigation of collaborative assessment of student teachers in Zimbabwe. In this model, both the college lecturer and a fellow student teacher are responsible for assessing a student teacher’s classroom practice. Alison Castro discusses her efforts to improve, through interventions in content and methods courses, elementary preservice teachers’ conceptions of mathematics curriculum materials. In an invited editorial, Lee Peng Yee reflects upon how he and other educators in Singapore are addressing the issue of mathematics subject knowlege in teacher training programs. In this issue’s ‘In Focus’ piece, Ginger Rhodes, Jeanette Phillips, Janet Tomlinson, and Martha Reems, who have worked together in mentoring student teachers, share two important ideas for improving the student teaching experience. In addition, they pose three questions that give teacher educators an opportunity to reflect upon their practice. The production of this issue is not a solo effort. I would like to thank the reviewers, authors, and faculty members whose work and advice have shaped the presentation of ideas found in the pages that follow. In particular, I extend my gratitude to our editorial and production staff, who has made this experience rewarding and memorable. On a personal note, I would like to take this opportunity to remember John Van de Walle, a colleague and leader within the mathematics education community, who recently passed away. John may be best known for his mathematics methods textbooks that are used in pre-service education programs across the United States and beyond its borders. He also authored numerous articles and chapters in NCTM publications and frequently shared his vision of mathematics education at conferences. John served in numerous capacities within NCTM, including as a director and as a committee member. It is through this latter capacity that I had the opportunity to meet and work with him, as we served together on NCTM’s Learning, Teaching, Curriculum, and Assessment Committee. During this time, I found John to be insightful and passionate in his work. I admired his leadership, vast wealth of knowledge, and sense of humor. As a mentor, his guidance and encouragement was invaluable as I transitioned from the high school classroom to graduate school. I, along with many others, will miss John and cherish fond memories of him. I am dedicating this issue in John’s memory and wish to extend condolences from us here at TME to his family, friends, and colleagues. Kyle T. Schultz 105 Aderhold Hall The University of Georgia Athens, GA 30602-7124

tme@uga.edu www.coe.uga.edu/tme

About the Cover This issue’s cover features a photograph of a stellated octahedron, an octahedron with a tetrahedron constructed on each of its faces. Joseph Harris, an Algebra 1 student at Clarke Central High School in Athens, Georgia, constructed this polyhedron from folded construction paper. A talented artist, Harris has constructed a number of polyhedra under the guidance of his teacher, Stephen Bismarck.

This publication is supported by the College of Education at The University of Georgia

____________ THE ________________ ___________ MATHEMATICS ________ ______________ EDUCATOR ____________ An Official Publication of The Mathematics Education Student Association The University of Georgia

Fall 2006

Volume 16 Number 2

Table of Contents 2 Guest Editorial… Mathematics for Teaching or Mathematics for Teachers? LEE PENG YEE 4 Uncovering Algebra: Sense Making and Property Noticing DAVID SLAVIT 14 Preparing Elementary Preservice Teachers to Use Mathematics Curriculum Materials ALISON M. CASTRO 25 Direct and Indirect Effects of Socioeconomic Status and Previous Mathematics Achievement on High School Advanced Mathematics Course Taking MEHMET A. OZTURK & KUSUM SINGH 35 Efficacy of College Lecturer and Student Peer Collaborative Assessment of InService Mathematics Student Teachers’ Teaching Practice Instruction LOVEMORE J. NYAUMWE & DAVID K. MTETWA 43 In Focus… Mentor Teachers’ Perspectives on Student Teaching GINGER A. RHODES, JEANETTE PHILLIPS, JANET TOMLINSON, MARTHA REEMS 47 Upcoming conferences 48 Submissions information 49 Subscription form

© 2006 Mathematics Education Student Association All Rights Reserved

The Mathematics Educator 2006, Vol. 16, No. 2, 2–3

Guest Editorial… Mathematics for Teaching or Mathematics for Teachers? Lee Peng Yee In Singapore, one topic of great interest in mathematics teacher education is the subject knowledge (SK) in mathematics of teachers. This topic has arisen in other places, often under the name “mathematical knowledge for teaching”, or “MKT.” Many mathematicians and mathematics educators believe that teachers should be given more content knowledge in mathematics. Such content knowledge should not be just higher mathematics but mathematics that is connected to mathematics teaching. In other words, it should be mathematics for teaching. Many books have been published on this subject in the United States during the past few years. McCrory (2006) provides a review of twenty such books. In addition, the Fall 2005 issue of American Educator features the topic of mathematics for teaching (for example, see Ball, Hill, & Bass, 2005). Issues At Singapore’s National Institute of Education (NIE)1, at least 8 years ago, we introduced a course on SK in mathematics for elementary school preservice teachers in one program. It has presently been extended to all three training programs for elementary preservice teachers. For the past few years, we have engaged in a discussion on what to teach in SK, how to teach it, and who should be teaching it. It is a controversial subject, at least among some mathematicians and mathematics educators in Singapore. I shall list the five issues currently under discussion. Issue 1: Is SK mathematics for teaching or mathematics for teachers? It is generally agreed that we should give teachers more mathematics than what they need. The question is how far we should go. For example, should we teach the topic of base-five numbers? Some say yes, some say no, and others a qualified yes, meaning just the concept and not technical aspects such as the four basic Lee Peng Yee has taught at universities in different countries for more than 40 years. He supervised more than 20 PhD students in mathematics. He was heavily involved in planning the school mathematics syllabus in Singapore and the degree program for teachers at the National Institute of Education, Singapore.

2

operations. The argument is whether SK should be a course on mathematics for teaching or mathematics for teachers. If it is the former, we cover only the topics taught in elementary school. If it is the latter, we should go beyond what is being taught in the classroom. The idea behind mathematics for teachers is not only to learn mathematics but also to learn enough mathematics so that teachers become more confident when teaching. Currently, the SK course being taught at NIE is meant to be a course on mathematics for teachers. One rationale is if we want our school pupils to attempt socalled challenging problems, then teachers themselves should have tried such problems. Furthermore, the problems should be challenging to teachers; that is, challenging at the level of teachers, not pupils. If teachers have never tackled problems challenging to them, how will they understand the difficulties that their pupils may encounter? Hence we must give them more mathematics beyond the elementary school. Issue 2: Is SK a course in mathematics or a course in mathematics education? By a course in mathematics, I mean the emphasis is on mathematics and rigor. By a course in mathematics education, I mean topics in the course may include problem solving heuristics. This does not mean that the two do not intersect. Of course, they do. The difference is in the approach. The mathematical approach may make use of heuristics implicitly, whereas the pedagogical approach may begin with heuristics and illustrate the heuristics using examples in mathematics. At NIE, it depends on who is teaching the course. Since the SK course at NIE is often taught by someone whose background is more mathematical than pedagogical, the approach naturally tends to be more mathematical. Issue 3: Who should be teaching the SK course? This issue is connected to Issues 1 and 2 above. Ideally, the SK course should be shared between a mathematician and a mathematics educator. In reality, it is often taught by one or the other and not jointly. The background training of the instructor seems to determine how the course will be taught. Though the Mathematics for Teaching or Mathematics for Teachers?

course outline may be the same, once inside the classroom, the lessons look different. Sometimes we wonder whether it is necessary for it to be taught in a uniform manner. Perhaps not.

Furthermore, the ancient Chinese introduced the concept of diameter before radius. Indeed, there is a word for diameter in Chinese, but not for radius. Radius in Chinese is simply called half-diameter.

Issue 4: Is rigor a major learning objective for the SK course?

Example 2

There are different possibilities for the degree of mathematical rigor used with preservice teachers. For example, should we use different notations for an angle and the measure of an angle? Should we distinguish between a line segment and the length of a line segment? Should we define an edge on a solid? Apparently, rigor means different things to different people. Some feel that we should be sufficiently rigorous when introducing concepts. At the same time, we want to keep the language dynamic and less rigid. For a mathematician, rigor comes before the Concrete, Pictorial, and Abstract teaching strategy (CPA).2 I guess the issue is not rigor, but the degree of rigor. Issue 5. Does every preservice elementary teacher need to take the SK course? Another way of stating the issue is whether some preservice teachers can be exempted from taking the course. Perhaps we can have a placement test to determine who does not need to take the SK course. Perhaps we should practice differentiated teaching, meaning we may teach different versions of SK to preservice teachers under different training programs. A course on subject knowledge It was a long intellectual discussion at NIE on what to teach in SK, and how to teach it. The course has evolved over the years to what it is today. It is generally agreed that we should have a course on SK in mathematics for elementary school teachers under training and a certain amount of rigor should be maintained. It is also agreed that we should make it refreshing so that students find it useful and have a greater motivation to learn. Let me give two examples to explain what I mean by refreshing. Example 1 The area of a circle can be given by the formula

! 2 d where d is the diameter of the circle. If we 4 take ! to be 3 then the inscribed circle of a square has A=

an area that is about 3/4 of the area of the square. This is more instructional than the formula A = ! r 2 . I saw a display of such a model in a department of mathematics at an old university in Rome. Lee Peng Yee

The inequality a 2 + b 2 ! 2ab can be verified algebraically and geometrically. An alternative form is (a + b) 2 ! 4ab . Consider a rectangle with area A and perimeter l . If the length and the width of the rectangle are respectively a and b, then A = ab and l = 2(a + b). Hence the above inequality becomes 2

'l$ % " ! 4 A . Fix l , then A is maximized when the &2# equality holds, that is, the rectangle is a square. Similarly, fix A and we can find the minimum l in terms of A. As we can see, the topics covered in the examples are school related. The presentation is accessible to teachers, and language used is familiar. The examples provide a glimpse of what I mean by mathematics for teachers. Conclusion I have written here what has been discussed at NIE on the subject of SK for mathematics teachers. While I may have personal views, I do not have answers for all the questions asked. In my view, the key factors in mathematics for teachers are “rigor” and “refreshing”. Our direction should be to document what we have done, to build up a closer link between mathematics and pedagogy, and to nourish a better understanding between mathematicians and mathematics educators. Hopefully, in time, SK will become an integrated part of teacher training. References Ball, D. L., Hill, H.C., & Bass, H. (2005, Fall). Knowing mathematics for teaching: Who knows mathematics well enough to teach third grade, and how can we decide? American Educator, 14–22, 43–46. McCrory, R. (2006). Mathematicians and mathematics textbooks for prospective elementary teachers. Notices of the American Mathematical Society, 53(1), 20–29. 1

The National Institute of Education is the sole teachertraining institute in Singapore. Aligned with initiatives set forth by Singapore’s Ministry of Education, NIE administers preservice, in-service, and graduate programs for teachers. 2

For more information on this strategy, see: Lee, P. Y. (Ed.). (2007). Teaching primary school mathematics, A resource book. Singapore: McGraw Hill. 3

The Mathematics Educator 2006, Vol. 16, No. 2, 4–13

Uncovering Algebra: Sense Making and Property Noticing David Slavit This paper articulates a perspective on learning to discuss ways in which students develop personal sense and negotiate meaning in a middle school algebra context. Building on a sociocultural perspective that incorporates mental objects, learning is described as a mutually dependent process involving personal sense making and the public negotiation of meaning. Analysis of student problem solving is focused on the development of taken-asshared meaning through an individual and collaborative analysis of the properties of various conceptual entities. The results suggest that functional properties inherent in linear relationships were more supportive in eliciting meaning making exchanges than were algebraic properties associated with generalized arithmetic, although the contextual nature of the linear tasks may have also supported the meaning making activity.

What is the difference between sense and meaning? Drawing on the philosophies of Vygotsky and Leont’ev, Wertsch (1991) distinguishes between sense and meaning by focusing on the personal and public aspects of activity. Lave, Murtaugh, and de la Rocha (1984) concur: “sense designates personal intent, as opposed to meaning, which is public, explicit, and literal (p. 73).” The personal act of reflection and the public act of communication relate in a manner that allows one’s personal reflections to mediate and be mediated by one’s interactions with the environment (Bauersfeld, 1992; Hiebert, 1992). From this perspective, sense is based on one’s individual reflections, whereas meaning has both a personal and public dimension. Throughout the paper, we will take the perspective that sense refers to the current status of cognitive acts constructed within an individual’s mental plane, and that an individual’s meaning is the inferred result of the intention or process of making one’s sense knowable within a social environment. In other words, one’s meaning is an intent to articulate one’s sense, but meaning is also a public, “taken-as-shared” (Cobb, Yackel, & Wood, 1992) construct developed in a social context. Because this paper focuses on learning and knowledge construction, we turn our attention Dr. David Slavit is Associate Professor of Mathematics Education at Washington State University Vancouver in the Department of Teaching & Learning and the Department of Mathematics. His research interests focus on preservice and inservice teacher development as well as student understandings of mathematical content, particularly algebra. Past Director of the Masters in Teaching program at WSUV, he has been principal investigator of numerous National Science Foundation grants, including the current Sustaining Teacher Research: Inquiry, Dialogue, and Engagement (STRIDE) research project. Dr. Slavit has also done extensive research on teaching and learning with instructional technology. Email: dslavit@wsu.edu 4

explicitly to the differences between sense making and meaning making, with the latter explored in a social dimension. The philosophical complexities in the analysis of collective meaning making, as opposed to an individual’s isolated meaning making, are greatly expanded due to the interactions between the personal and added social dynamics at play. For example, a radical constructivist perspective emphasizes the interpretations of one’s experiences in framing realties, so the distinctions between sense making and meaning making become quite complex. As Lerman (1996) states: Rejecting a picture theory of mind, that mental representations of reality are exact replicas of the real world, leads, for the radical constructivist, to the conclusion that one can only argue that all representations are constructed by the individual, and hence meanings are ultimately those in the individual’s mental plane. (p. 137)

The cognizing individual constructs her or his own world (i.e., makes sense) out of the articulated meanings put forth by others in social interactions. But the meaning making of others is filtered by the organization of an individual’s own experiences and sense-making processes, and any attempt to convey one’s meaning is, in turn, filtered by the sense-making processes of others (von Glaserfeld, 1995). Thus, sense and meaning are quite blurred, as the result of attempts at collective meaning making are always unknown across the individuals’ own mental planes. One’s own meaning, as well as the perceived meaning of others, must remain in the internal realm of sense making. Others, however, draw more separation between the notions of sense and meaning. For example, many researchers have articulated a view of learning in which “mental objects” are constructed through reflections on actions and activity. This mental Uncovering Algebra

abstraction has been called reification (Sfard and Linchevski, 1994), encapsulation (Dubinsky, 2000), and verb-noun status (Davis, 1984). For these researchers, sense making leads to distinct acts of meaning making where individuals construct and publicly share clear descriptions of these mental objects. Positioned perhaps between the above two approaches, Cobb et al. (1992) describe a sociocultural perspective of learning in which students make use of experiences and interactions, mediated by developmental interventions of the instructor, in order to construct understandings that are either principally generated by one’s sense-making activities or that approximate and expand on the taken-as-shared meanings of a community or society. This approach goes beyond the acquisition and participation metaphors of learning and includes aspects of knowledge creation (Paavola, Lipponen, and Hakkarainen, 2004). Here, it is possible that the approximation of others’ meaning in an individual’s sense-making activity can be thought to be negligible. Slavit (1997) has built on the above perspectives to articulate a theory involving the development of meaning in the context of algebraic ideas through an awareness of the properties of publicly negotiated and taken-as-shared mental objects. The ability to make sense of the properties that are embedded within and help define a situation, activity, symbol system, or idea can lead to the development of richer forms of senseand meaning-making activity. These developments commonly occur in the cognitive act of formalization (Kieren, 1994), which allows an individual to make sense of mathematical constructs. For example, students over time might recognize that linear functions grow at a constant rate (from a numeric or graphic perspective), are continuous, and have exactly one x- and y-intercept (excepting vertical and horizontal lines). These and other properties lead to an understanding of linear functions borne from the sensemaking process. Further, these understandings can then be weighed against one’s perceptions of the taken-asshared meanings of society as a whole, including negotiation with one’s peers; these understandings could also be initially generated out of interactions with one’s peers. In this perspective, the learner develops mental constructs associated with established mathematical objects and ideas by focusing on his or her interpretation and awareness of the properties that define these objects and ideas.

David Slavit

Therefore, constructing sense and meaning in algebra can be approached by focusing on the properties associated with the objects and ideas that help define this specific area of mathematics. Although algebra is a multi-faceted mathematical area, this study is primarily concerned with the two areas defined as “generalizing and formalizing patterns and constraints” and “study[ing] of functions, relations, and joint variation” (Kaput, 1995). Patterns in arithmetic computations can be identified by noting properties that these computations share, which can then be generalized to formal algebraic rules.1 These generalizations form the basis of understanding algebra as generalized arithmetic and illustrate the development of sense and meaning in this area of algebra through a property-noticing process. Similarly, one can experience growth relationships between two varying quantities in a variety of situations to support the development of more abstract notions of functions, including the concept of covariation (a patterned change in one variable due to a patterned change in another; Kieran & Sfard, 1999; Slavit, 1997). Method The purpose of the study is to examine the sensemaking and meaning-making activities of pairs of students engaged in problem-solving episodes related to the algebraic topics of generalized arithmetic and function. While not attempting to prove that collaborative learning environments are more effective than individual settings, the study investigates the manner in which pairs of students use specific cognitive and social processes in algebraic problemsolving environments. Data consist of 15 videotaped interviews involving problem-solving episodes with fourteen 7th grade and sixteen 8th grade students from two middle school classrooms in a rural, mediumsocioeconomic status school in the northwestern United States. The students worked in pairs on two tasks (detailed in Figure 1) for approximately 20–30 minutes. Approximately half of the students in each of the two classrooms were randomly selected to participate, and all interviews were transcribed. The eighth grade students had a limited amount of formal exposure to algebra prior to the study, consisting mainly of equation solving, whereas the seventh grade students had no previous formal algebra instruction. Although whole-class discussion of problem solving strategies and solutions were common in both classrooms, the students had limited prior exposure to working in collaborative pairs.

5

Task 1) Two carnivals are coming to town. You and your friend decide to go to different carnivals. The carnival that you attend charges $10 to get in and an additional $2 for each ride. The carnival your friend attends charges $6 to get in, but each additional ride costs $3. If the two of you spent the same amount of money, how many rides could each of you have ridden?

. . Task 2) What digit is in the one’s place of the number: 29 34 56 Figure 1. Problem-solving tasks used to measure students’ sense of functional algebra and generalized arithmetic, respectively.

The interviews were conducted in a small area of a quiet room, and the students were under no time constraints to complete the tasks. The students were given one copy of the first task and were told to “work the problem anyway you wish, but you may wish to work together.” Only pencil and paper were provided. When it seemed that the students were at the end of their solution attempt, they were questioned on the manner in which they approached the task, and asked to think about other ways to solve the problem. This looking back stage was intended to promote critical reflection by the students on their solution strategy. These procedures were repeated for the second task. Analysis was conducted on the videotaped segments as well as the students’ written work. The analysis centered on the kinds of algebraic understandings (Kaput, 1995) the students seemed to bring into the problem-solving situation, and the kinds of algebraic understandings that the pairs utilized and constructed in their solution. Hence, the social, external, and mathematical constraints inherent in problem solving were present, but the analysis centered on the construction of understandings. Particular attention was given to the kinds of properties that the students attached to the algebraic ideas and mental objects that helped support their investigation. Numerous researchers have extended a sociocultural view of learning to research on teaching, grounded in participant’s actions and perspectives (Cobb et al., 2003; Simon and Tzur, 1999). Researcher participation, and the methodology itself, can be jointly negotiated with the participants. Kieren et al. (1995) took a similar approach to research on learning, describing an enactive learning environment that attempts to balance the overall aspects of the learning situation with the cognitive and social backgrounds of the participants as they engage in mathematical activity. Kieren et al. used a group interview format where the mathematical activity and research focus are mediated by researchers, participants, and setting. Kieren and his colleagues prefer to balance the role of one’s sense-making activity and one’s ability to 6

negotiate and construct meaning. As they state, instead of situated cognition or situated cognition, research should focus on situated cognition. Hence, analysis of student mathematical activity should be concerned with understanding the students’ individual cognitive processes in the context of the entire setting, including the genesis and nature of interactions that lead to knowledge construction. Likewise, a discussion of the setting should be framed by the activity that occurs within. This perspective was the lens through which the interviews in this study were both conducted and analyzed. Discussion of results will focus on overall trends in the data, followed by a microanalysis of problem-solving interactions. Results Task 1 was designed to allow students to approach the problem either arithmetically or from a more algebraic perspective. Students can solve the problem by simply adding the cost of a ride or rides to each admission and finding the amount of money where

Person 10 12 14 16 18 20 22 24 26 28 30

A

Person 6 9 12 15 18 21 24 27 30 33 36

B

Number of Rides 0 1 2 3 4 5 6 7 8 9 10

Figure 2: Amounts of money spent by carnival attendees for a given number of rides. Uncovering Algebra

these totals are the same (see Figure 2). However, students could also extend this strategy and find many other combinations where the amount of money spent is the same. Specifically, the students could express their answer in a manner that describes a variable relationship between the amount of rides and money spent. Students could discuss the general relationships in the numeric solution pairs that were obtained and then set the expressions 10 + 2a and 6 + 3b equal to each other, or they could draw the graph of this linear relationship. These strategies would suggest a more formal algebraic approach to the task than the computational method described above because of the greater degrees of abstraction present in the representation and solution process. Task 2 could be worked in a purely computational way, where the value of 29 ⋅ 34 ⋅ 56 is computed and the value of the one’s place identified. However, students could also identify a factor of 10 present in the product, either through initial computation or by an examination of the factors, and recognize that the value of the product must end in zero. Task 1 Overall analysis revealed that sense-making and property-noticing activities of some students occurred in this setting, which may not have been constructed if working alone. However, slightly less than half of the pairs of students worked almost independently of one another, and very little knowledge was shared. Students who worked collaboratively constructed solution strategies that involved meanings introduced by both students, and the congruencies and incongruencies in the individual strategies seemed to advance the collective approach in most, but not all, working pairs (see Table 1). The solution of Tim and Molly2, discussed later, provides a clear example of a pair of students whose collective solution strategy yielded results that may not have evolved if working alone. Every group began the problem by listing dollar amounts for a given number of rides, and every group but one found either the solution (one ride for Person A, two rides for Person B, $12) or the solution (4, 4, $18). Overall, more than half of the student pairs attended to covariance properties in regard to the number of rides or money spent (see Table 1). This analysis either involved the relationship between the increase in the number of rides of each carnival attendee for every new solution (three additional rides for Person A and two additional rides for Person B), or

David Slavit

an increase in the amount of money spent for every new solution (one new solution for every $6 increase). A greater number of eighth grade pairs than seventh grade pairs performed an analysis of the rate at which new solutions were found in regard to money spent (see Table 1). This result was due to the fact that the seventh grade student pairs were more likely to focus only on situations in which both riders went on the same number of rides. Finding the amount of money each person would spend for the same number of rides does not include an analysis of the times when each spends the same amount of money but enjoys a different number of rides. Students who only focused on a uniform increase in the number of rides across riders found the solution (4, 4, $18), but failed to find solutions such as (1, 2, $12). Students investigating Task 1 in ways that did not involve a uniform increase in the number of rides had to perform a complex analysis of the covariance properties present in the situation. The students first had to recognize that the two quantities relating to the amounts of money spent by each attendee increase at different rates as more rides are taken. Then the students had to realize that this difference impacts another form of covariance—each person’s number of rides is different for each solution. The latter difference is due to a 3:2 ratio between the number of rides of the two attendees from one equal spending value to the next (see Figure 2). Therefore, to make these kinds of generalizations, the students had to simultaneously negotiate two different covariance situations and relate them to their analysis. The eighth grade students were more likely to discuss a general solution to the task by finding a pattern in the solutions displayed in a numeric table of values they created. An example of this kind of analysis is provided below. Many of these groups took a more formal approach by making explicit notice of the manner of covariation in the number of rides and money spent, as just described. These kinds of observations are pivotal when attempting to generalize a solution for an arbitrary number of rides. Two other groups made explicit mention of covariation that was inappropriate to the problem context, and four groups made no explicit mention of any notion of covariance. One pair of students explored these notions using algebraic symbols, but the rest of the students did not go beyond the numeric table and verbal descriptions.

7

Table 1 Algebraic understandings of covariance and solution possibilities exhibited by students on Task 1 No. groups 7th grade (7 groups) 3

8th grade (8 groups) 3

Made explicit notice of covariation in regard to amount spent (a new solution for every 6 dollar increase)

1

4

Made no explicit mention of any notion of covariance

3

1

Stated or suggested that they realized that their were a theoretically infinite number of solutions to the task

3

4

Stated or suggested that they realized that there were “a lot” of solutions to the task

2

4

Found only one solution and did not seem to think others could be found

2

0

Group’s algebraic understanding Made explicit notice of the differences in covariation in the number of rides and money spent between the two carnival attendees (3:2 increase in number of rides per additional solution)

Analyzing the property of covariance (i.e., slope) from multiple situational perspectives eluded many of the student pairs, most of whom were in the seventh grade (see Table 1). Overall, some general differences existed in the level of algebraic reasoning and awareness of algebraic properties between the students in the two grade levels. This may have been due to the eighth grade students’ prior exposure to algebraic ideas, including notions of variable and equation solving. Task 2 Collaborative problem-solving behaviors by the student pairs on Task 2 occurred less frequently than on Task 1. All but two of the student pairs either worked in isolation, with each student computing the value of the product individually, or with no real collaborative problem-solving activity. The latter involved negotiation of individual computational duties, such as one person calculating 29 and the other person finding the values of the other two factors. All students who worked in isolation compared their answers to the calculations of their partner. Overall, there was very little interaction between the students on this task. The solution strategies on Task 2 were also more uniform. Only one seventh-grade pair and one eighthgrade pair who successfully answered the question did not calculate the value of the expression. These two student pairs noted the presence of the factors two and 8

five in the product, recalled the property of a factor of 10 producing a zero in the one’s place, and then utilized this property to construct a solution. However, unlike Task 1, simple probing questions, such as “Is that the only way you could have done it?”, elicited generalizations in many of the student pairings who had already solved the problem through direct calculation. In particular, the realization of the presence of a factor of 10 led to the development of strategies in three of the seventh grade pairs and three of the eighth grade pairs similar to that provided by the two pairs of students mentioned above. Therefore, this task did not initially elicit algebraic solution strategies, but these behaviors did arise when prompted by the researcher. Enactive Learning: A Closer View This section will focus on the problem-solving strategies of Tim and Molly, two eighth graders, as they worked together on Task 1. It will explore the contextual, social, and cognitive factors that may have played a role in their problem-solving processes. The transcription of the solution strategy developed by Tim and Molly illustrates how they initially constructed an understanding of the problem based on the properties of the two functions that represent the amounts of money spent by the two carnival attendees. These properties involved the initial amount (admission) and rate of increase (cost per ride). Analysis centered on their individual sense-making and collective meaningUncovering Algebra

making activities as they engaged in a solution attempt. This interview was chosen because of the degree of collaboration and collective sense making that occurred. Although shared meaning developed between Tim and Molly, the previously discussed data indicates that such activity was not found in all of the interviews. After Tim found an initial solution of one ride and two rides (1, 2, $12), Molly conjectured that more solutions would be possible. They discussed the covarying properties of the two dependent variables, and this led Tim to the solution (4, 4, $18). At this point, the two participants realized that infinitely many solutions were possible, but they had no means of describing what these solutions would be. Therefore, the students approached the problem from a computational perspective by making use of the functional notion of covariance to find alternate solutions, but they were unable to generalize to the arbitrary case. Hence, their solution showed aspects of algebraic thought, but they failed to develop functional properties associated with arbitrary quantities. Instead, the idea of covariance was contemplated from an arithmetical perspective. Further, the students’ ability to articulate their sense-making processes led to additional meaning making between the pair, leading to the acquisition of higher degrees of understanding of the mathematical situation. A closer look at specific portions of the interview reveals particular instances of sense making and instances where the students noticed properties that led to their solution. The following segment occurred at the onset of their solution attempt on Task 1:

multiple, almost, or something like that, but anyway. Molly: You actually (gibingly). Tim:

learned

something

That wasn’t funny (good-humoredly).

Molly: One of my family’s jokes.

The beginning portion of the interview illustrates several important aspects present throughout the interview. First, the two students felt comfortable with one another and did not appear to be nervous or affected by aspects of the interview setting (e.g., the video camera, presence of researcher, pressure of solving the problem). Second, the students were individually engaged in the task and actively sought an understanding of the context and solution strategy. The next few lines of the interview illustrate that the two were beginning to make use of each other’s sense-making activities. Molly: This actually depends on how many rides you went on, to go on, if you wanted to go on like two rides, you could spend Tim:

It depends, no, OK

Molly: for each Tim:

OK, if I wanted, if I was here and you there, right, if I wanted to go on two rides that would be a total of 14 dollars, for me, if you wanted to go on two rides then it would only be 12 dollars for you, so it would end up costing

T & M: (read problem, mumbling)

Molly: Oh, I was mixed up, OK (laughs) well, one of them, one person didn’t have to ride at all to get 10 dollars, no

Molly: I don’t know.

Tim:

Tim:

Wow (exasperated). (pause) Do you have a calculator?

DS:

No.

Tim:

OK, so right now we know that each person paid at least 10 and 6 dollars.

Molly: And then they could also go on two times, but it depends how many rides he’d want to go to. Tim:

How many rides can each—

Molly: Actually, it depends on how many rides he’d want to go on. Tim:

David Slavit

No actually, um, actually it’s pretty much asking what, like, sort of asking, it’s almost like asking lowest common

OK, so what we are trying to figure out is, they spent the same amount

Molly: Yeah I know, OK, and six dollars, what, what adds up to being, lets see, 10, 20, (long pause), OK

Molly began to explore the problem by advancing on her initial sense of the situation, which involved an understanding of the need to consider “how many rides.” She made use of the cost of admission and ride price, situational properties that correspond to the linear functional properties of y-intercept and slope. They were beginning to construct and make use of shared meanings of the situation. Tim utilized Molly’s remarks regarding the need to consider the case of going on two rides to begin his numerical analysis. After these computations, Molly recognized that this would not be a desired solution 9

and said, “I was mixed up.” In the last two comments, they returned to their individual sense-making activities. Tim then made the following key insight: Tim:

I could go on one and you could go on two and then we’d each spend 12 dollars (very confidently). That’s the answer (chuckles). And 12 dollars is one answer, or, ‘cause

Molly: Yeah that’s true. Tim:

You’d go twice (writes) 12

Molly: So you’d Tim:

No, one person would go. Yeah (put pencil down) I get it, there’s that one.

Using the covariance properties previously explored, Tim decided to vary the number of rides for each person, changing the (2, 2) case to (1, 2). This produced a solution that Molly was able to immediately verbalize. Tim’s personal solution became a shared and meaningful one, although Tim was clearly the initiator of nearly all of the public meaning. But this immediately changed when I became part of the interaction after assuming from Tim’s last comment that the problem-solving activity had ended: DS:

You spent 12? OK, so the one was two and the other was one ride?

Tim:

(writes) 6 times 30

DS:

Or what makes you say that?

Tim:

and if the other person goes on four, or wait a minute, this person goes on two.

Molly: It would be the same amount just as the first time. DS:

By the first time you mean when the one person rides one and the other person rides two?

Molly: OK, one person Tim:

No it wouldn’t (confidently). OK, wait, so you’re saying, so you’re saying

Molly: OK, you go Tim:

one person goes four times, goes on four rides

T & M: and the other person goes on two Tim:

that’s 12, 14, this one’s 9, 12, 15, 18, so, no, that’s not exactly true.

Molly: What (contentiously)? Tim:

You said one person goes on two and another person goes on four.

Molly: So, but wait. Tim:

‘cause this person has to pay six just to get in and three for each ride

Molly: Well you could spend more time and

Molly: yeah

Tim:

Tim:

You could spend a lot more money, and then

Molly: Yeah Tim:

You could

Molly: One person could go on two rides and the other person could go on four rides and you’d still get the same thing.

Molly’s current sense of the solution allowed her to expand the situation by linearly increasing their solutions, with some minor prompting from me. However, Molly’s generalization was inaccurate, as it did not include a proper analysis of the rates of increase in the number of rides between solutions. While Molly was verbalizing the meanings she constructed that led to this conjecture, Tim makes 10

OK, what would they spend in that case?

Molly: Well, if you had

So

Molly: one person would go on two rides and the other person would go on one. Tim:

DS:

I’d go once,

Molly: You’d go once Tim:

sense of Molly’s remarks and challenges the meaning put forth by Molly after conducting a few computations:

that’s four rides, that’s a total of 12 right there, plus another six is 18, so that’s not necessarily true.

Molly: Well, um, if you take, it’s 10 and six, and then the first time, one person goes on one once Tim:

that’s two dollars right there

Molly: yes, that’s two dollars, and then another person goes on another time, that’s two times, and then you go on it again, six, and another two, I guess that doesn’t work. It’s worked in the past for me.

Using the meanings put forth by Tim, Molly recognized the faults in her sense of the situation and altered her belief in her own solution of (2, 4). The pair Uncovering Algebra

constructed a collective sense of the precise nature of the covariation in the situation and used this to further their understanding and solution of the problem, including a greater awareness of the property of covariance. Eventually, their meaning making exchanges collectively led to the (4, 4) solution, but either participant made no further generalizations. However, Tim does suggest that he believes more solutions are possible: Tim:

You could find other answers if you’d keep going but, if you had all the time in the world I’m sure you could find a lot more answers.

Molly: If you were an old fogey I bet you could, because you’d have a lot of time.

Despite failing to advance the solution, the above dialogue illustrates how the articulation of sense and joint construction of meaning are interdependent processes that can lead to knowledge acquisition. Molly and Tim successfully completed Task 2, but their analysis was not as thorough and did not employ any algebraic methods. Their solution strategy involved performing the entire multiplicative computation stated in the problem, and then examining the digit in the one’s place in their final answer. The pair shared the computation and writing duties throughout, but did not construct a solution in a truly collaborative manner. Molly: It’s two to the ninth, so one more (multiplication of two). Tim:

OK, that’s 512. Here, you do this for a while. (hands Molly the paper)

Molly: OK.

After the pair completed the computation, I initiated a discussion that did not generate a more algebraic approach, with Tim concluding by stating, “I think there’s a shorter way but we don’t know.” The use of the term “we” suggests that Tim was viewing the pair as a fully functioning team throughout this task as well, even though no real collaboration occurred. However, unlike Task 1, the pair was unable to utilize algebraic methods in this solution because the ability to identify and make use of the effect of multiplication by 10 was not apparent. As discussed before, this result was typical across all but two of the groups without prompting from the researcher. Analysis of the work of another eighth-grade pair provided a different perspective on the nature of collaboration and the use of algebraic properties. Like Tim and Molly, Bill and Gloria collaboratively David Slavit

explored Task 1 and arrived at multiple solutions and a detailed analysis of the 3:2 ratio that existed between the increase in rides. On Task 2, both students initially asked for calculators, and then divided up the computation. Bill:

I’ll do two to the ninth and you do three to the fourth.

Gloria: OK.

The pair continued computing silently for a very long time, with both eventually working on 56 together. However, before this computation was completed, Gloria stated, “Wouldn’t we get zero?” Bill either ignored or dismissed this comment and continued his computation. After several more minutes, the pair arrived at an answer of zero. Gloria: Zero will be there, and if you multiply anything by zero you get zero. DS:

Can you look at the problem and see why?

Bill:

(long pause) Not really.

Gloria:

We got zero here (at one point in the computation) and it stayed zero after that.

Although Bill and Gloria began working collaboratively on the computational aspect of the problem, they did not share in any meaning-making activity regarding the conceptual aspects embedded in this task. Gloria began to explore this, but did not progress either individually or collaboratively with Bill. As stated above, this lack of group meaning making was true of the majority of paired groupings on this task. Conclusion In many situations, learning is a collective process of privately constructed personal sense and publicly negotiated meaning. The ability to utilize one’s sense to articulate meaning, as well as make sense of other’s stated meanings, enriches the taken-as-shared network being constructed. Learning is a dynamic interplay between one’s sense, one’s stated meanings, and the sense one makes out of other’s stated meanings. This study provides evidence where these three aspects of learning can combine to form a collective, rich understanding of a problem-solving situation, or a situation where sense- and meaning-making do not fully develop. This study does not intend to make the case that students are more successful working together. Rather, it tries to articulate how specific conceptual processes are utilized when working on algebraic tasks in both an individual and collaborative environment, and what 11

kinds of algebraic tasks might elicit individual and collaborative problem-solving behavior. The main mathematical object of analysis in Task 1 was the linear growth relationship that contained the property of constant rate of change and a covariance property between the two variables. In the case of Tim and Molly, the taken-as-shared meanings that developed were certainly a product of the two individual’s sensemaking activities, but the development of these meanings also effected future sense-making and meaning-making activities in the task. These students were both willing and able to participate in this personal and shared process, and their collective senseand meaning-making activities were at the heart of their learning experiences. Overall, the tasks and learning environment were able to elicit sense-making activities based upon the individual students’ understandings that were then transferred into meaning-making activities shared by the student pairs. These activities led to solution strategies that involved the generalization of arithmetic constructs into more algebraic realms. This occurred in the context of interactions between students in Task 1, but interactions with the researcher also led to these occurrences in Task 2 for several student pairs. The most significant advancements made by the student pairs appeared to have occurred in Task 1, where the students explored aspects of the covariance to advance their understandings of the task, the number of solutions found, and their ability to articulate a generalized solution. The fact that there was more interaction on Task 1 may have been due to the fact that the linear growth properties inherent in this task were more apparent to the students than the multiplicative or number theory properties inherent in Task 2. The use of a context involving two people may have also made Task 1 easier to model than Task 2. As a result, the students were able to participate in richer meaning making exchanges about Task 1, and develop more advanced solutions. It appears that, for these middle school students, algebraic understandings of the property of covariance on Task 1 were readily available. These understandings allowed many of the student pairs to approach the task in algebraic realms, making use of the properties of covariance, correspondence, and slope to identify multiple solutions to the task. Understandings of appropriate arithmetic properties which may have led to a more generalized approach to Task 2 were not as apparent. Hence, in this study, the students were better able to utilize their sense-making activities to uncover properties related to aspects of functional algebra than 12

with properties associated with algebra as generalized arithmetic. But the limited scope of this study does not begin to allow for generalizations of this result. Further research is needed to discuss differences in students’ facility with various algebraic properties and the ability to work with these properties at various levels of generality (Kaput, 1995). Moreover, students with procedural problemsolving tendencies (which many of the students appeared to possess) would not be expected to utilize algebraic understandings on Task 2 because a solution strategy requiring direct computation is immediately recognized. But, in Task 1, a student who begins to compute the number of rides associated with various amounts of money spent by the carnival attendees is generating information that can lead to a discovery of notions of slope and covariance, which happened frequently in these student pairs. Therefore, the problem-solving behaviors of the students may have led to differences in their ability to recognize properties of the mathematical situations and to solve problems with various degrees of generality across the two tasks. These data illustrate the complexities inherent in the personal and public interplay of students’ knowledge construction processes that often go unnoticed in an interactive, problem-solving environment. References Bauersfeld, H. (1992). Classroom cultures from a social constructivist’s perspective. Educational Studies in Mathematics, 23, 467–481. Cobb, P., Yackel, E., & Wood, T. (1992). A constructivist alternative to the representational view of mind in mathematics education. Journal For Research In Mathematics Education, 23(1), 3–31. Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Schauble, L. (2003). Design experiments in educational research. Educational Researcher, 32(1), 9–13. Davis, R. B. (1984). Learning mathematics: The cognitive science approach to mathematics education. London: Routledge. Dubinsky, E. (2000). Mathematical literacy and abstraction in the 21st century. School Science and Mathematics, 100(6), 289– 297. Hiebert, J. (1992). Reflection and communication: Cognitive considerations in school mathematics reform. International Journal of Educational Research, 17(5), 439–456. Kaput, J. J. (1995). A research base supporting long term algebra reform? In D. T. Owens, M. K. Reed, & G. M. Millsaps (Eds.), Proceedings of PME-17 (Vol. 1), (pp. 71–94). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education. Kieran, C. & Sfard, A. (1999). Seeing through symbols: The case of equivalent expressions. Focus on Learning Problems in Mathematics, 21(1), 1–17.

Uncovering Algebra

Kieren, T. E. (1994). Bonuses of understanding mathematical understanding. In D. F. Robitaille, D. H. Wheeler, & C. Kieran (Eds.), Selected lectures from the 7th International Congress on Mathematical Education (pp. 211–228). SainteFoy, Quebec: Les Presses de L’Universite’ Laval. Kieren, T. E., Calvert, L. G., Reid, D. A., & Simmt, E. (1995, April). Coemergence: Four enactive portraits of mathematical activity. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco. Lave, J., Murtaugh, M., & de la Rocha, O. (1984). The dialectical construction of arithmetic in grocery shopping. In B. Rogoff & J. Lave (Eds.), Everyday cognition: Its development in social context (pp. 67–94). Cambridge, MA: Harvard University Press. Lerman, S. (1996). Intersubjectivity in mathematics learning: A challenge to the radical constructivist paradigm? Journal for Research in Mathematics Education, 27(2), 133–150. Paavola, S., Lipponen, L., & Hakkarainen, K. (2004). Models of innovative knowledge communities and three metaphors of learning. Review of Educational Research, 74(4), 557–576 . Sfard, A. & Linchevski, L. (1994). The gains and pitfalls of reification: The case of algebra. Educational Studies in Mathematics, 26, 191–228.

David Slavit

Simon, M. & Tzur, R. (1999). Explicating the teacher's perspective from the researchers' perspectives: Generating accounts of mathematics teachers' practice. Journal for Research in Mathematics Education, 30(3), 252–264. Slavit, D. (1997). An alternate route to the reification of function. Educational Studies in Mathematics, 33(3), 259–282. von Glaserfeld, E. (1995). Radical constructivism: A way of knowing and learning. London: RoutledgeFalmer. Wertsch, J. V. (1991). Voices of the mind: A sociocultural approach to mediated action. Cambridge, MA: Harvard University Press.

1

For example, a2 - b2 = (a + b)(a - b) expresses the relationship between the difference of the squares of any two numbers and the product of their sum and difference. An additional example involves the discovery that the result of any product that contains a factor of 10 must have a zero in the one’s place.. 2

All participant names are pseudonyms.

13

The Mathematics Educator 2006, Vol. 16, No. 2, 14–24

Preparing Elementary Preservice Teachers to Use Mathematics Curriculum Materials Alison M. Castro Learning how to use mathematics curriculum materials to create learning opportunities is an important part of the work of teaching. This paper presents findings from a study involving 15 elementary preservice teachers enrolled in, first, a content and, then, a methods course, and discusses the extent to which three curriculum interventions influenced their conceptions of how math curriculum materials are used. Additionally, this paper discusses the implications of this research for mathematics teacher education programs and proposes a framework for integrating work around curriculum materials into mathematics content and methods courses in order to prepare preservice teachers for using these materials effectively.

Recent efforts by the National Council of Teachers of Mathematics (NCTM) to improve the way that K-12 mathematics is taught and learned have implications for mathematics teacher education. As teacher education programs aim to develop teachers’ knowledge of mathematics and their knowledge of students as learners, these programs “should [also] develop teachers’ knowledge of and ability to use and evaluate instructional materials and resources” (NCTM, 1989, p. 151). In particular, “teachers need a well-developed framework for identifying and assessing instructional materials…and for learning to use these resources effectively in their instruction” (p. 151). Although using mathematics curriculum materials effectively is an important part of teachers’ work, it is an aspect of practice that is often overlooked in teacher education programs. Prevalent in classrooms across the country, curriculum materials are important and can be influential resources for teachers. Mathematics curriculum materials, in particular, are potentially influential given the challenging nature of mathematics instruction espoused under recent reform efforts. In response to NCTM’s (2000) recommendations regarding the improvement of mathematics instruction, some mathematics curriculum materials have become highly designed and very detailed sources of both Dr. Alison Castro is Assistant Professor of Mathematics Education and Learning Sciences at the University of Illinois at Chicago where she also serves as the Research Director for the Teaching Integrated Mathematics and Science (TIMS) Project, which investigates the impact of the Math Trailblazers elementary curriculum on teachers’ practice and student understanding in the classroom. Her research interests focus on preservice and in-service teachers’ use of mathematics curriculum materials to create opportunities to learn and the factors that influence this use.

14

content and pedagogical information (Trafton, Reys, & Wasman, 2001). From homework and grouping suggestions to examples of student errors and alternative solution strategies, such innovative mathematics curricula provide a potential wealth of information and instructional support for teachers. Given the quantity of information and pedagogical suggestions in these innovative materials, however, teachers can potentially use mathematics curriculum materials in a number of different ways. Whereassome teachers tend to follow the suggestions in mathematics curriculum materials almost as a script for instruction (Graybeal & Stodolsky, 1987; McCutcheon, 1981), other teachers do not rely on the suggestions in teacher materials to the same extent, but rather adapt the suggestions and activities as they see fit (Stake & Easley, 1978). Furthermore, it is possible that teachers use curriculum materials to only a limited extent, if at all. The increasing use of innovative mathematics curricula combined with the aforementioned research raises concerns about teachers’ use of curriculum materials—that teachers can use these materials with an inattention to the actual content and nature of the tasks, activities, and pedagogical suggestions contained in these resources. Research illustrates that the moves and decisions of teachers during instruction influence the nature of students’ work, often reducing the complexity and challenge of the tasks and activities (Stein & Lane, 1996; Stein, Smith, Henningsen, & Silver, 2000). Given that mathematical tasks lie at the center of innovative curriculum materials, teachers’ perceptions and use of these materials can potentially have a strong influence on how these tasks are enacted. If teachers are inattentive to the nature of the given tasks in the curriculum materials and their enactment of these tasks, they can undermine the task complexity. Preparing Elementary Preservice Teachers

On the other hand, if teachers are deliberate and purposeful in their use of curriculum materials to inform their moves and decisions around tasks, they may be better able to maintain students’ engagement in complex, intellectual mathematical work. Given the need for teachers to be informed users of mathematics curriculum materials, it is important to understand how teachers think about and learn to use these materials. Preservice teachers’ conceptions of and experiences with such materials provide insight into this process. Little has been written about how preservice elementary teachers learn to use mathematics curriculum materials. Ball and FeimanNemser (1988) found that, during student teaching, preservice teachers varied in their use of these materials. The researchers attributed this differentiated use to what students’ teacher education programs were advocating about mathematics curriculum materials. However, little more is known about how preservice teachers learn to use curriculum materials and even less is known about how teacher education programs develop preservice teachers’ skills at using these resources effectively. Thus, this study aims to (a) explore preservice elementary teachers’ conceptions of mathematics curriculum materials, (b) analyze what and how preservice teachers learned from activities that were designed to help them learn to use mathematics curriculum materials effectively, and (c) discuss the implications of this research for preservice teacher education programs.1 Conceptual Framework Teachers can use curriculum materials in a number of different ways. Some teachers rely heavily on curriculum materials, following the suggestions in an almost prescriptive manner (Remillard, 1992), while other teachers modify and adapt curriculum materials in the course of planning for and teaching a lesson (Stake & Easley 1978). Other researchers have explored different factors that impact teachers’ use of curriculum materials. From policy guidelines (Floden et al., 1980; Kuhs & Freeman, 1979) and teachers’ interpretations of policies (Cohen et al., 1990) to teachers’ ideas about the purpose of education and nature of learning (Donovan, 1983; Stephens, 1982), teachers’ use of curriculum materials is influenced by a number of different factors. Although the research on teachers’ use of curriculum materials and the factors influencing this use primarily focus on inservice teachers, the extant research can be used as a framework for understanding how preservice teachers interpret and use these Alison M. Castro

materials. Just as inservice teachers draw upon resources when making decisions about curriculum materials, preservice teachers rely on similar types of resources to make decisions about their practice. Addressing the different resources that teachers draw upon when teaching, Cohen, Raudenbush, & Ball (2002) highlight teachers’ knowledge, skill, and will. In their framework, resources such as teachers’ knowledge and skill can influence how curriculum materials are utilized. “For when teachers…use resources, they make judgments about which to use, how to use them, with whom, and to what end. They base these judgments on what they know and believe about themselves, one another, and the content” (p. 104). However, resources such as knowledge and skill may be limited for preservice teachers. Due to their inexperience, preservice teachers may have a narrow view of teaching and classroom practice, and limited or incomplete conceptions of the ways in which curriculum materials can be utilized. Moreover, preservice teachers bring preconceptions about teaching into their teacher education programs (Lampert & Ball, 1998). They have spent years as students in the classroom watching their teachers and developing ideas about good teaching, but they know little about the decisions and challenges teachers actually face in the classroom. In particular, preservice teachers are most likely unaware of the ways in which teachers use mathematics curriculum materials to make decisions. As preservice teachers have very limited personal resources to draw upon when making teaching decisions, teacher education programs can aim at influencing such resources. That is, elementary education programs can develop preservice teachers’ knowledge and skill at using mathematics curriculum materials in ways that develop and further students’ understanding. As Cohen et al. (2002) describe, Though some teachers judge with great care and seek evidence with which they might revise, others are less careful. In either event, teachers calibrate instruction to their views of their capacities and their students’ abilities and their will to learn. (p. 104) Indeed, Simon & Schifter (1993) reported that teacher education programs in mathematics education can help teachers develop a conception of teaching and learning that is consistent with recent reform ideas. And Ball (1990) argues that mathematics methods courses, in particular, can influence preservice teachers’ knowledge, assumptions, and beliefs about mathematics. Methods and content courses, taken 15

together, are a potentially useful venue for providing preservice elementary teachers with the necessary knowledge, skills, and conceptions of mathematics curriculum materials to enable them to use these materials effectively.2 Methodology This study took place in the context of two required courses in a year-long Master’s and certification program in Elementary Education at a large, midwestern university. The first course, a mathematics content course for elementary teachers, was an 8-week summer course designed to prepare preservice teachers for teaching the core elementary mathematical domains of number and operations. The second course in this sequence was an elementary mathematics methods course, which took place the following semester.3 Notably, much of the underlying philosophy of these two courses drew on NCTM’s (1989, 2000) recommendations regarding the teaching and learning of mathematics, including activities with manipulatives that drew heavily on NCTM’s recommendations regarding the use of these materials. Fifteen students volunteered to participate in this study. Because this study solicited participants on a volunteer basis, this is not a random sample, and thus, may not be representative of the population of preservice elementary teachers at this institution. Many of the students were considered non-traditional college students in that they left full-time employment in order to enroll in the Master’s program. In addition, several of the students had a variety of informal teaching experiences prior to entering the program, though it was not required for admission. Students’ notebooks comprise the first data source for this study. As part of their grades for each course, students were required to keep a notebook. These notebooks provided a space for them to write any inclass notes and annotations of course readings, as well as to complete weekly assignments. In addition to recording students’ ideas and responses to the curriculum activities, students’ notebooks also provided a space for them to answer questions related to their conceptions of mathematics curriculum materials at the start of the content course (referred to hereafter as pre-sequence).4 Individual interviews comprise the second data source for this study. Using semi-structured interviews, students were interviewed once following the completion of the content course (referred to hereafter as mid-sequence), and again following the completion of the methods course (referred to hereafter as post16

sequence). During both of these interviews, students were asked about their conceptions of mathematics curriculum materials, the perceived role of these materials in the classroom, and their thinking about the curriculum activities. The interview protocols for both sets of interviews included, among other questions, the same questions that were asked at the pre-sequence time point. See Appendix A for the mid-sequence interview protocol.5 Both the student notebooks and interview transcripts were analyzed and coded. From these codes, analytic documents were compiled, which were then used to generate inferences, and eventually hypotheses, to understand how students thought about math curriculum materials and the curriculum activities (Erickson, 1985; Strauss & Corbin, 1985). Curriculum Activities Over the two courses, three interventions were designed and administered to students to help them develop their capacities to use mathematics curriculum materials in deliberate and skillful ways. These curriculum activities were designed to draw specific attention to the nature and extent of the mathematical and pedagogical information in curriculum materials. Activity 1 took place during the third week of the content course, and comprised about 60 minutes of class time. The purpose of this activity was to first determine the mathematical goals of a textbook lesson in order to better understand how the lesson fit into the larger unit, and then to determine the mathematics that children were expected to learn from the lesson. In particular, students were asked to read a specific textbook lesson from a teacher’s guide and think about the following questions: What is this lesson about? What is the mathematics that children are supposed to be learning? What do the problems or exercises seem to intend children to do?6 Activity 2 took place during the fourth week of the content course, in which students were discussing mathematical proofs and methods of proving. This activity comprised about 60 minutes of class time. Specifically, students were discussing proofs of statements involving even and odd numbers. They were presented with four different definitions of even numbers taken from four different elementary mathematics curriculum programs. The purpose of this activity was to first understand the different definitions, and then to determine which types of numbers would be considered even given the definitions. For each definition, students were asked to (a) determine whether it is mathematically valid, (b) discuss whether it would be usable by third graders, Preparing Elementary Preservice Teachers

and, in the case that the definition is not usable, (c) revise the definition to make it appropriate for third graders. Finally, Activity 3 took place during the eighth week of the methods course, in which students were beginning to discuss the elements of lesson planning. This activity comprised about 75 minutes of class time. The purpose of this activity was to analyze a textbook lesson with careful attention to not only the larger lesson goals, but also to the other elements of the lesson, such as the tasks, examples, language, problem contexts, and mathematical representations.7 The underlying rationale was that, by analyzing the sequence of tasks, the language used, and the different mathematical representations included, students would be better able to make informed decisions about instruction and then modify the lesson where necessary. In short, these interventions were designed to help students learn different aspects of mathematics curriculum material use. These interventions also provided opportunities for students to focus on the mathematical and pedagogical aspects of a lesson, and make determinations and assessments of the nature and extent of the information accordingly.8

materials. Their responses were grouped into the following categories: textual materials, which includes teacher guides, transparencies, assessment resources, textbooks, student notebooks and journals, etc.; nontextual materials, which includes pencils, paper, calculators, and other materials or items that can be used by students during a mathematics lesson; and manipulatives, which includes Base 10 blocks, pattern blocks, Unifix cubes, and any other commercially made materials, as well as any teacher-made manipulatives.9 Table 1 displays students’ responses. As Table 1 illustrates, throughout the two courses, students had varied conceptions as to what constitutes mathematics curriculum materials. Some students considered mathematics curriculum materials to be exclusively textual materials, non-textual materials, or manipulatives. Other students considered these materials to be some combination of the three different categories. Although students’ conceptions of curriculum materials consistently fell into these three categories at pre-, mid-, and post-sequence, the distribution of their conceptions changed. The number of students who included non-textual materials as curriculum materials decreased from pre- to midsequence, and then again from mid- to post-sequence. As these numbers decreased, the number of students citing some combination of textual materials and manipulatives increased throughout the two courses.

Analysis of Students’ Understandings of Math Curriculum Materials This section is divided into three parts. To understand how students’ conceptions changed over time, the first part describes what items and materials students considered to be mathematics curriculum materials. The second part discusses how students envisioned these different items and materials being utilized in the classroom. Finally, the third part examines how students understood the three curriculum activities.

How Can Mathematics Curriculum Materials Be Used? In addition to being asked what constitutes mathematics curriculum materials, students were also asked to describe how these materials can be used in the classroom. Students’ responses varied along two dimensions. While some students thought that curriculum materials could be used to help students learn, others saw these materials as tools that can support teachers’ instructional decisions.

What Constitutes Mathematics Curriculum Materials? The students in this study considered a variety of materials and resources as mathematics curriculum Table 1

Students’ Conceptions of Mathematics Curriculum Materials Textual Materials Presequence Midsequence Postsequence

Alison M. Castro

Manipulatives

Textuals & Manipulatives

2

Nontextual Materials 0

Manipulatives & Non-textuals

3

Textuals & Nontextuals 2

2

Textuals, Non-textuals, Manipulatives 5

1

4

0

1

5

1

0

4

3

0

2

6

1

1

2

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Table 2 Students’ Conceptions of How Curriculum Materials Can Be Used Pre-sequence Mid-sequence Post-sequence

Help children learn 9 2 0

As Table 2 illustrates, students’ conceptions of curriculum materials varied along two primary dimensions. While at mid-sequence some students saw these materials as being used by children, others viewed curriculum materials as tools that support teachers’ decisions. By post-sequence, students’ views shifted, and all fifteen students viewed mathematics curriculum materials as supporting teachers’ decisions. In addition, students said that teachers can use these materials in one of three different ways: (a) scripted use, where a teacher relies heavily on the materials; (b) modified use, where a teacher modifies or adapts curriculum materials as they see fit; and (c) limited use, where a teacher uses the curriculum materials to only a limited extent. Although students did not appear to possess these views of curriculum use at presequence, these three distinct views emerged at midand post-sequence. At post-sequence, more students said that curriculum materials can be adapted and that teachers do not necessarily have to, as one student stated, “follow exactly what is stated in the book.” Influence of Curriculum Activities Throughout the study, students were asked specifically about the three curriculum activities and the extent to which they found the different activities useful in learning how to use mathematics curriculum materials. Table 3 displays students’ responses. At mid-sequence, students were asked specifically about Activity 1. In response to this question, only six students specifically mentioned this activity as useful in learning how to use mathematics curriculum materials. Whereas two of these students said Activity 1 was useful insofar as it introduced and exposed them Table 3 Students’ Perceptions of Utility of Curriculum Activities Student Learning Curriculum Activity 1 2 3 18

Supported 6 4 12

Not Supported 2 1 0

No mention 7 10 3

Support teachers 4 13 15

Both 2 0 0

to different mathematics curricula, the other four students cited this activity as particularly useful to their learning. One of these four students stated the following: I think it was a helpful activity because maybe as a new teacher you would just kind of, oh well this is the teacher’s guide, this is how I need to teach this. But by thoroughly examining it and you know, looking at the math that’s going on and … maybe you would see the faults in the book.

All four of these students also had similar conceptions of how curriculum materials can be used. Specifically, they said that teachers should modify and adapt curriculum materials in order to meet the needs of their particular classrooms. In contrast to these six students, two of the remaining nine students specifically stated that this activity was not useful for learning how to use curriculum materials because they felt too inexperienced to make such decisions. In particular, one student said, “I found it very difficult to get access to. And I think I also thought of myself at that point in the course as someone who didn’t really have the background to decide whether the math was good enough.” Similarly, the other student said that he did not have enough experience to summarize the mathematics in a lesson. Thus, these students felt they lacked the background and experience to properly summarize the mathematical ideas in curriculum materials. At mid-sequence, students were also asked specifically about Activity 2. As with Activity 1, this activity was mentioned by only a few students. Specifically, four students mentioned that this activity was helpful for learning how to use mathematics curriculum materials. Three of these students stated that this activity helped them to think about definitions in textbooks differently than they had before. “I can see some of the deficits of stuff that’s out there…. For instance, I don’t think, from what I’ve seen so far, the [name of a curriculum program] is not particularly strong in definitions.” Another student stated that she

Preparing Elementary Preservice Teachers

found this activity particularly useful because it helped her to think about mathematics in a different way. And I think that you can’t just look at—I think some teachers take this teacher guide and that’s what they teach and this is what they talk about versus looking in it and saying well what else do we have about that … Because sometimes people get caught up with reading what’s in front of them and that’s how it is … It was eye-opening what we did.

In contrast to these four students, one student said that they were not experienced enough to evaluate mathematical definitions in curriculum materials. It is not clear whether the inexperience felt by this student was due to a lack of strong content knowledge or to the overall design of the activity. Finally, at post-sequence, students were asked about the extent to which Activity 3 helped them learn how to use mathematics curriculum materials. In response to this question, an overwhelming number of the participating students viewed this activity as not only a useful activity to include in such a course, but as an important part of teachers’ work in the classroom as well. In contrast to the first two curriculum activities, twelve of the students in this study found this activity very useful for learning how to use curriculum materials. For example, when asked about the utility of this activity, one student replied as follows: I had to think about … different angles … that helped me look at the curriculum in a very detailed, deliberate way … I needed to go back and see what they [referring to children] had done before, needed to see what they were going to do next.

Similarly, another student stated the following: I think it was useful because I don’t know that I looked at the lesson plans so critically before … it’s kind of one of those things that you just took their word for it, if it was in the book you should teach it … I got the message [from Activity 3] that you take the lesson plan and kind of alter it according to your students.

Notably, eight of these twelve students adhered to the same view of curriculum use—that teachers should modify and adapt curriculum materials according to children’s different abilities. In short, students generally saw Activity 3 as useful for learning how to use mathematics curriculum materials. Furthermore, they felt that it was applicable to teachers’ daily practice. However, only a small proportion of the students in this study viewed Activities 1 and 2 as helpful for learning how to use

Alison M. Castro

curriculum materials, and important to their own learning as future teachers. Students’ perceptions of the utility of a curriculum activity seemed markedly related to how they viewed curriculum use—several students who cited Activity 3 as useful to their learning thought that teachers should appropriately modify and adapt curriculum materials for their classroom. Discussion As their conceptions of curriculum materials shifted throughout the two courses, students formulated and reformulated ideas about the particular ways in which teachers can use mathematics curriculum materials. Ranging from strict use to modified use to no use at all, students seemed to have very clear notions of the different ways in which teachers can and should use curriculum materials. Generally, these findings indicate that across the two courses, the focus of students’ conceptions of mathematics curriculum materials shifted. Although many students focused at pre-sequence on how children interact with curriculum materials, that focus changed by post-sequence to highlight and include teachers’ interactions with these materials. The shift in students’ conceptions of what constitutes mathematics curriculum materials clearly illustrates their shift to a knowledge-based conception of curriculum materials. At pre-sequence, several students considered non-textual materials such as paper, pencils, and rulers to be curriculum materials. Although these materials can be used by children in a variety of activities, both during and not during instruction, they do not directly support children’s learning of mathematics. At post-sequence, students’ conceptions of curriculum materials focused more heavily on textual materials and manipulatives. Textual materials and manipulatives are both predominately used during or in preparation for instruction, and in comparison to non-textual materials, textual materials and manipulatives are more directly involved with children’s acquisition of knowledge. Thus, over the course of the sequence, students’ conceptions of mathematics curriculum materials seemed to have shifted to a more knowledge-based conception. This shift in students’ conceptions of curriculum materials also indicates a move to a more teacherbased conception of these materials. At pre-sequence, students’ views of curriculum materials focused primarily on children’s use of these materials. At postsequence, however, students’ views of curriculum materials primarily focused on how teachers can use such materials. This shift makes sense in light of the

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shift described in the above paragraph. Knowledgebased curriculum materials support teachers’ planning and instruction more directly (and perhaps to a greater extent) than non-textual materials. Knowledge- and teacher-based materials can be used to embody mathematical content, which is precisely what is taught during a lesson. As students moved towards a more knowledgeand teacher-based conception of curriculum materials, they began formulating particular views of how teachers can use these materials. At pre-sequence, students’ responses did not clearly indicate that they had considered how teachers can use such materials. By mid-sequence, students had formulated somewhat concrete views of curriculum use—scripted use, modified use, and limited use.10 Most students thought teachers should modify and adapt curriculum materials. By post-sequence, these newly formulated ideas had even begun to shift as more students thought that teachers should modify these materials. So, although students held different views at mid- and post-sequence regarding teachers’ use of curriculum materials, students almost universally began to develop their conception of such use over the span of the two courses. This trend comports well with the shift in students’ conceptions of what constitutes curriculum materials described above. As students began to consider curriculum materials as tools that support instruction, they also began to consider how teachers use such tools to inform their teaching. From the research conducted in this study, it is not clear why students formulated their particular views of how teachers should use curriculum materials. That is, it is unclear why many students believed that teachers should modify and adapt mathematics curriculum materials, whereas other students believed teachers should either strictly use these materials or not use them at all. Students certainly received messages about curriculum materials from their cooperating teachers and other sources external to the two courses, such as periodicals. Moreover, students may have received implicit (and explicit) messages about how to use curriculum materials from these two courses.11 Nevertheless, the findings indicate that students were in fact formulating concrete views of how teachers should use curriculum materials during the two courses. Moreover, the precise effect of the curriculum activities on students’ conceptions of curriculum materials and how these materials can be used is not immediately evident. Students indicated that the first two activities were not very useful. However, students 20

generally viewed Activity 3 as a practical activity that teachers would do on a regular basis. Activity 3 is designed to help students focus on the mathematical content embodied in curriculum materials, such as textual materials and manipulatives. The finding that students found this activity useful comports with students’ knowledge- and teacher-based conceptions of curriculum materials. Students considered the activity useful precisely because it could support instruction. Although on one hand it is possible that the lesson analysis activity contributed to students’ conceptions of curriculum materials and curriculum use, it is also possible that students found the lesson analysis useful because it supported their already existing conceptions. In all likelihood, both of these possibilities are simultaneously true. During the two courses, students were constantly formulating and reformulating their conceptions. The effect of any given curriculum activity, in part, depends on students’ conceptions of curriculum use before the activity begins. Implications In short, the trends that emerged in this study indicate that students moved to more teacher-based conceptions of what constitutes mathematics curriculum materials and how these materials can be used. Also, students’ conceptions of curriculum materials shifted over the two courses to include more knowledge-based materials, such as teacher’s guides, assessment resources, and manipulatives. Despite these changes in students’ conceptions of curriculum materials, it is unclear to what extent these changes can be attributed to the curriculum activities. Moreover, as the curriculum activities did not directly influence students’ conceptions, it is unclear whether the two courses together impacted how students thought about using mathematics curriculum materials. However, mathematics content and methods courses are able to provide students with at least some conceptions of curriculum materials to enable them to use these materials in skillful ways. To be sure, it seems unreasonable to think that three curriculum activities will equip students with all of the necessary skills to enable them to use curriculum materials effectively. The curriculum activities in this study did not seem to influence students’ conceptions of curriculum materials nor did these activities broaden students’ potentially limited resources, as described by Cohen et al. (2002). However, what is evident is that students’ coursework can, in part, influence such resources. For this reason, further work needs to be done to create a more cohesive framework for

Preparing Elementary Preservice Teachers

mathematics content and methods courses that integrates curriculum materials into the coursework to a greater extent. Such a framework for content and methods courses should include several components that are crucial to helping students learn to use curriculum materials effectively. Table 4 displays the different components of this framework. First, content and methods courses should expose students to different mathematics curricula and provide opportunities for students to learn about and familiarize themselves with the potential resources that are available to them. When describing their thoughts about the different curriculum activities, three students in this study stated that these activities were only useful to them insofar as the activities exposed them to different mathematics curricula. Second, students should develop a discriminating eye towards math curricula. That is, students should have opportunities to look across an entire program; assess what information is provided for teachers, how the lessons are structured over the school year, and how the various curricular components are related; and also to evaluate the extent to which the program is aligned with different standards and frameworks (when applicable). Third, students should have opportunities to select, develop, and possibly adapt mathematical tasks and appropriate instructional strategies that are typically provided in curriculum materials. Although mathematical tasks are important to children’s learning, the work teachers do with tasks is even more important. Teachers’ decisions and actions influence the nature and extent of children’s engagement with challenging tasks, and ultimately affect children’s opportunities to learn (Stein et al., 2000). Students need to learn to assess the difficulty of mathematical tasks provided in curriculum materials in order to implement tasks appropriate for children’s current mathematical ability, and then, when necessary, modify or adapt tasks in ways that maintain the integrity of the task. Also, students should learn to determine whether the given instructional suggestions are appropriate, and, if not, to identify and employ instructional strategies that will better facilitate children’s learning. Another important element of students’ coursework is to consider the use of manipulatives. Throughout the two courses, more than half of the students included manipulatives in their conceptions of mathematics curriculum materials. Moreover, as the students in this study described how manipulatives can be used in the classroom, a majority of these students stated that manipulatives can not only be used to Alison M. Castro

Table 4 Framework for Mathematics Content and Methods Courses Component Exposure to curriculum materials

Purpose Expose students to potential curricular resources they may use in the future.

Developing discriminating eye

Help students develop an overall understanding of a math curriculum program (what is important, valuable, and needs to be modified) and recognize alignment with state standards and curriculum frameworks.

a

Math task analysis

Help students select, develop, and possibly adapt tasks in ways that maintain task integrity; identify appropriate instructional strategies.

Effective manipulative use

Help students use manipulatives in ways that support and maintain children’s understandings of concepts.

accommodate children’s different abilities, but also to make mathematics fun for and applicable to children. As noted by several researchers, manipulatives can often be used in unsystematic and unproductive ways (Ball, 1992; Moyer, 2001; Stein & Bovalino, 2001). Although teachers may have well-designed lessons incorporating manipulative-based tasks, children’s work may not automatically develop in ways that support their understanding of the mathematics (Stein & Bovalino, 2001). In addition, children often learn to use manipulatives in a rote fashion, with little emphasis and understanding of the mathematical concepts behind the procedures (Hiebert & Wearne, 1992). Thus, students need to learn to use manipulatives that support and scaffold children’s learning, as opposed to simply making mathematics fun and applicable to children’s everyday lives, as mentioned by several students in this study. By redesigning mathematics content and methods courses to prepare prospective teachers to use these resources effectively in their instruction, we can enable future teachers to more effectively provide students with a high quality education. If prospective teachers were better prepared to use mathematics curriculum materials to create learning opportunities for students, they would potentially be better prepared to manage the complexities of teaching. While [new] teachers may not be able to act upon such [curriculum] knowledge immediately, it gives them a mindset to inform their deliberations about teaching, to view the issues of classroom … in a 21

larger context, and to be dissatisfied with the compromises and survival tactics of the first year as they continually reassess their own teaching in an attempt to provide an appropriate learning environment for their students. (Zumwalt, 1989, p.182)

By designing mathematics content and methods courses that prepare preservice teachers to use curriculum materials, we are preparing them to become knowledgeable professionals that are part of a larger community of educators. In addition to outlining a framework for preservice programs, this study raises important issues that should be taken into consideration when integrating curriculum material-related coursework into content and methods courses. First, the findings draw attention to the influential role of students’ cooperating teachers, as several students mentioned their cooperating teacher in their field placement when describing how mathematics curriculum materials can be used. Some students seemed to be influenced by what they saw and heard from their cooperating teachers in their field placement, a phenomenon identified by other researchers (Ball & Feiman-Nemser, 1988). Thus, it is certainly possible that some students receive messages regarding curriculum materials from cooperating teachers that are inattentive to the nature of the contents and suggestions in curriculum materials. At the same time, they simultaneously receive conflicting messages from their teacher education programs that promote careful and deliberate use of these materials. Although it is not clear how to respond to such a situation, it is important to be aware of any external and opposing influences on students’ coursework. In closing, this study raises several important issues related to preservice teachers’ conceptions and use of mathematics curriculum materials. By understanding the conceptions and assumptions preservice teachers bring to teacher education programs about mathematics curriculum materials, teacher educators can become better able to design coursework and implement activities that will help students learn to use these materials in skillful ways. References Ball, D. (1990). Breaking with experience in learning to teach mathematics: The role of a preservice methods course. For the Learning of Mathematics, 10(2), 10-16. Ball, D. (1992). Magical hopes: Manipulatives and the reform of math education. American Educator, 16(2), 14-18, 46-47.

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Ball, D., & Feiman-Nemser, S. (1988). Using textbooks and teachers' guides: A dilemma for beginning teachers and teacher educators. Curriculum Inquiry, 18, 401-423. Cohen, D., Raudenbush, S., & Ball, D. (2002). Resources, instruction, and research. In F. Mosteller & R. Boruch (Eds.), Evidence matters: Randomized trials in education research (pp. 80-119). Washington DC: Brookings Institution Press. Cohen, D., Peterson, P., Wilson, S., Ball, D., Putnam, R., Prawat, R., et al. (1990). The effects of state-level reform of elementary mathematics curriculum on classroom practice (Final Report to OERI – Elementary Subjects Center Series No. 25). East Lansing, MI: Michigan State University, Center for Learning and Teaching of Elementary Subjects. (ERIC Document Reproduction Service No. ED323098) Donovan, B. (1983). Power and curriculum in implementation: A case study of an innovative mathematics program. Unpublished doctoral dissertation, University of Wisconsin, Madison. Floden, R., Porter, A., Schmidt, W., Freeman, D., & Schwille, J. (1980). Responses to curriculum pressures: A policy-capturing study of teacher decisions about content. Journal of Educational Psychology, 73, 129-141. Erickson, F. (1985). Qualitative methods in research on teaching (Occasional Paper No. 81). East Lansing, MI: Michigan State University, Institute for Research on Teaching. (ERIC Document Reproduction Service No. ED263203) Graybeal, S., & Stodolsky, S. (1986, April). Instructional practice in fifth-grade math and social studies: An analysis of teacher’s guides. Paper presented at the annual meeting of the American Educational Research Association, Washington D.C. (ERIC Document Reproduction Service No. ED276614) Hiebert, J., Wearne, D. (1992). Links between teaching and learning place value with understanding in first grade. Journal for Research in Mathematics Education, 23, 98-122. Kuhs, T., & Freeman, D. (1979, April). The potential influence of textbooks on teachers' selection of content for elementary school mathematics (Research Series No. 48). Paper presented at the annual meeting of the American Educational Research Association, San Francisco, CA. (ERIC Document Reproduction Service No. ED175856) Lampert, M., & Ball, D. (1998). Teaching, multimedia and mathematics. New York, NY: Teachers College Press. McCutcheon, G. (1981). Elementary school teachers’ planning for social studies and other subjects. Theory and Research in Social Education, 9, 45-66. Moyer, P. (2001). Are we having fun yet? How teachers use manipulatives to teach mathematics. Educational Studies in Mathematics, 47, 175-197. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Author. Remillard, J. (1992). Teaching mathematics for understanding: A fifth-grade teacher's interpretation of mathematics policy. The Elementary School Journal, 93, 179–183. Remillard, J. (1996). Changing texts, teachers, and teaching: The role of curriculum materials in mathematics education reform. Unpublished doctoral dissertation, Michigan State University, East Lansing, MI. Preparing Elementary Preservice Teachers

Remillard, J. (2004). Teachers' orientations toward mathematics curriculum materials: Implications for teacher learning. Journal for Research in Mathematics Education, 35, 352-388. Remillard, J. (1999). Curriculum materials in mathematics education reform: A framework for examining teachers' curriculum development. Curriculum Inquiry, 23, 315-342. Simon, M., & Schifter, D. (1993). Towards a constructivist perspective: The impact of a mathematics teacher inservice program on students. Educational Studies in Mathematics, 25, 331-340. Stake, R., & Easley, J. (1978). Case studies in science education. Urbana: University of Illinois. Stein, M., & Bovalino, J. (2001). Manipulatives: One piece of the puzzle. Mathematics Teaching in the Middle School, 6, 356359. Stein, M., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2(1), 50-80. Stein, M., Smith, M., Henningsen, M., & Silver, E. (2000). Implementing standards-based mathematics instruction. New York: Teachers College Press. Stephens, W. (1982). Mathematical knowledge and schoolwork: A case study of the teaching of developing mathematical processes. Unpublished doctoral dissertation, University of Wisconsin, Madison. Strauss, A., & Corbin, J. (1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage. Trafton, P., Reys, B., & Wasman, D. (2001). Standards-based mathematics curriculum materials: A phrase in search of a definition. Phi Delta Kappan, 83, 259-264. Zumwalt, K. (1989). Beginning professional teachers: The need for a curricular vision of teaching. In M. C. Reynolds (Ed.), Knowledge base for the beginning teacher (pp. 173-184). Oxford, England: Pergamon Press.

1

Given the nature and focus of this study, it is important to define what is meant by curriculum, as this term often has multiple meanings. In some cases, curriculum refers to the overarching national, state, and district-level frameworks that specify what is to be taught in classrooms. Curriculum can also refer to the resources teachers use to plan for and guide their instruction. For the purposes of this study, curriculum refers only to the resources used by teachers. Consequently, curriculum materials refer to items such as teacher guides, assessment resources, manipulatives, and any other materials that accompany a particular mathematics curriculum program.

ways intended by curriculum developers per se. Instead, the author argues that students can learn how to discriminately use these resources to select, develop, and/or adapt the features within these materials, such as mathematical tasks and suggested instructional strategies, in order to create effective learning opportunities for students. 3

The author was the instructor for the content course and was not the instructor for the methods course. 4

The primary reason for soliciting studentsâ€™ initial conceptions of math curriculum materials in this fashion was that it may have been uncomfortable for them to be interviewed upon immediately starting the program. 5

The mid- and post-sequence interview protocols include the same subset of questions. The questions regarding the curriculum materials are similar across both protocols, but the wording is specific to the curriculum activities in question when appropriate. 6

Students were given the same textbook lesson.

7

Students were given the same three textbook lessons.

8

It is important to note that the different textbook lessons used in the curriculum activities came from both nonStandards-based and Standards-based mathematics curricula. The latter, in this case, refer to curricula that were supported by the National Science Foundation (NSF) funds in the early 1990s that were commissioned to create mathematics programs that were aligned with the ideas put forth by the NCTM Standards (1989). 9

Although, arguably, some of the materials included as non-textual materials (e.g., rulers, protractors, calculators, etc.) could be considered manipulatives, the materials listed as manipulatives were considered to be (and were used in both courses as) manipulatives. Furthermore, the non-textual materials listed were not considered manipulatives in these courses. These categories were identified based on studentsâ€™ responses and the different materials that were used and discussed in both courses. 10

These three categories of curriculum use comport with inservice teachersâ€™ use of curriculum materials as found by Remillard (2004). 11

As the instructor for the content course, the author supported a modified view of curriculum use. That is, using curriculum materials in a modified or adaptive fashion. However, it is not clear how the instructor for the methods course discussed curriculum use.

2

It is important to note that the author is not implying that students should learn how to use curriculum materials in the

Alison M. Castro

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Appendix A Mid-sequence Interview Protocol Introduction: As you know, I am conducting a study of preservice elementary teachersâ€™ views of mathematics curriculum materials. In this interview, I will be asking you questions about how you think about mathematics curriculum materials in general, the role they play in the classroom, and how teachers can use these materials to help students learn mathematics. Finally, I will be asking you questions about the curriculum materials activities from your class this semester. Mathematics Curriculum Materials: 1) What do you think of when you hear the phrase â€œmathematics curriculum materials?â€? (Issue is how respondent defines what constitutes mathematics curriculum materials) 2) Why do you think of [list items, ideas mentioned by respondent in previous question] when you think of mathematics curriculum materials? 3) What role do you think mathematics curriculum materials play in the classroom? 4) Why do you think so? 5) In what ways do you think teachers can use mathematics curriculum materials to help students learn mathematics? (Issue is how respondent thinks mathematics curriculum materials can and should be used in the classroom, regardless of their experience with these resources) EDUC 518 Class: 1) Throughout your class this semester, you talked about and engaged in activities that were directly related to mathematics curriculum materials. In particular, you worked on analyzing a textbook lesson, which included examining the tasks, examples, language, representations, as well as the overall mathematical ideas embedded in the lesson. What are your thoughts about this activity? If respondent asks for clarification: Did you find this activity useful or not useful? If so, in what ways? 2) Over the course of this semester, what do you think you learned about mathematics curriculum materials? 3) How do you think you learned about mathematics curriculum materials? 4) Were there other activities or discussions, either in this course or in Math 485, that you think helped you learn about mathematics curriculum materials? (Issue is whether respondent thinks of other activities from either course that impacted how they think about mathematics curriculum materials) Conclusion: 1) I really appreciate you taking the time to talk with me. Is there anything else you would like to add to what we have already talked about here? 2) Do you have any questions for me before we finish this interview?

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Preparing Elementary Preservice Teachers

The Mathematics Educator 2006, Vol. 16, No. 2, 25–34

Direct and Indirect Effects of Socioeconomic Status and Previous Mathematics Achievement on High School Advanced Mathematics Course Taking Mehmet A. Ozturk Kusum Singh Direct and indirect effects of socioeconomic status (SES) and previous mathematics achievement on high school advanced mathematics course taking were explored. Structural equation modeling was carried out on data from the National Educational Longitudinal Study: 1988 database. The two variables were placed in a model together with the mediating variables of parental involvement, educational aspirations of peers, student’s educational aspirations, and mathematics self-concept. A nonsignificant direct effect of SES on course taking suggests the lack of an ‘automatic’ privilege of high-SES students in terms of course placements. The significant indirect effect of previous mathematics achievement tells that it needs to be translated into high educational aspirations and a strong mathematics self-concept to eventually lead to advanced course taking.

In view of the freedom students in public high schools have in choosing the courses they take, course taking is a critical aspect of the students’ education. Research consistently suggests that this is especially true for mathematics. Course taking was found to have the largest effect on academic achievement in mathematics among the academic subjects examined (e.g., Jones, Davenport, Bryson, Bekhuis, & Zwick, 1986; Schmidt, 1983). Researchers found that, even when students’ social background and previous academic achievement were controlled, course taking was the single best predictor—twice as strong as any other factor—of achievement in mathematics (Lee, Burkam, Chow-Hoy, Smerdon, & Geverdt, 1998). In a study on National Educational Longitudinal Study: 1988 (NELS: 88) data, results indicated that the achievement growth differences in mathematics and science among high- and low-socioeconomic status students completing the same numbers of courses were small. This was particularly true in mathematics, where none of the socioeconomic status (SES) comparisons were significant among students taking the same numbers of high school mathematics courses (Hoffer, Mehmet A. Ozturk is an assistant professor of educational research at Cleveland State University. His research interests include secondary school mathematics and science achievement, psychosocial and educational consequences of adolescent parttime work, and academic expectations in elementary and secondary schools. Kusum Singh is a Professor of Educational Research and Evaluation at Virginia Tech. Her research focuses on family, social and school factors that affect academic achievement, particularly in mathematics and science. She teaches graduate courses in statistics and research methods. Mehmet K. Ozturk & Kusum Singh

Rasinski, & Moore, 1995). Researchers hypothesize that mathematics is almost exclusively learned in school and that background factors do not exert much of an influence through out-of-school learning (Jones et al.; Schmidt). A parallel interpretation would be that much of the SES differences in mathematics achievement gains over the high school grades are due to the different numbers of mathematics courses that high- and low-SES students complete during high school (Hoffer et al.). Advanced level mathematics bears importance for almost all students, regardless of their plans for the future. For college admissions and success, these courses play a critical role. Data collected from students admitted to four-year colleges and universities show the high numbers of advanced mathematics courses completed by these students (Owings, Madigan, & Daniel, 1998; U.S. Department of Education, 1997). For students who want to enter the job market after high school, these courses are also beneficial. Many jobs that once required little knowledge of mathematics now call for various skills in algebra and measurement. According to an industrywide standard, an entry-level automotive worker should have the knowledge to apply formulas from algebra and physics in order to properly wire the electrical circuits of any car (U.S. Department of Education). These advanced courses have also been shown to improve students’ fundamental mathematical skills such as problem solving (Jones, 1985; Rock & Pollack, 1995). Two fundamental background variables that are related to advanced mathematics course taking are SES 25

and previous mathematics achievement. SES has been a significant variable in studies that looked at equity in advanced course taking. There has been a concern regarding any possible discrimination in course placements based on students’ SES or minority status (e.g., Calabrese, 1989; Lareau, 1987; Useem, 1991). It may be that parents with higher income and educational levels are at an advantage in terms of being able to support their children toward advanced courses. This can happen in a number of legitimate ways such as role modeling and providing help with homework. However, the intriguing question is whether or not higher levels of SES bring about an ‘automatic’ privilege in course placements. Any such practice by school administrators or teachers, such as allowing parents of higher SES to have a greater say in their children’s course placements, would be a form of discrimination. Previous mathematics achievement is also closely related to advanced level mathematics course taking, since it has almost always been a consideration in advanced mathematics course placements. The critical question about previous mathematics achievement is whether students who have succeeded in prerequisite courses automatically enroll in more advanced courses or whether such achievement, by itself, is not enough for these students to further their studies in advanced mathematics. For example, there are studies suggesting that many students will not proceed with advanced courses—despite their proven record—unless they are encouraged to do so or feel confident about their mathematical abilities (e.g., Lantz & Smith, 1981). Almost none of the mathematics courses that would be classified as advanced are required for graduation, and around 60 percent of high school students graduate without having taken any of these advanced courses (Council of Chief State School Officers, 2002; Finn, Gerber, & Wang, 2002). Given these circumstances, it is important to translate previous achievement into advanced course taking. Literature Review SES and Advanced Mathematics Course Taking The term socioeconomic status is used by sociologists to denote an individual or family’s overall rank in the social and economic hierarchy (Mayer & Jencks, 1989). In most research, including national studies, SES has been measured as a combination of parents’ education, parents’ occupational prestige, and family income (Mayer & Jencks; White, 1982). Researchers have explored a number of mechanisms through which SES exerts its influence on 26

course taking. The first one is, undoubtedly, role modeling. Students from middle or high SES families constantly see, in their parents and neighbors, social and economic payoffs that good education could provide, while many minority children in high-poverty areas have few—if any—role models who have succeeded in school or who have translated school success into economic gain (Lippman et al., 1996; Oakes, 1990). These claims suggest an indirect effect of SES on advanced course taking through students’ educational aspirations. Another mechanism in the relationship between SES and advanced mathematics course taking is the high educational expectations middle- and high-SES families have for their children (Khattri, Riley, & Kane, 1997; Muller & Kerbow, 1993), and the conveyance of these expectations to them (Anderson, 1980; Hossler & Stage, 1992; Seginer, 1983; Wilson & Wilson, 1992). Lantz and Smith (1981) found significant positive relationships of educational aspirations and parental encouragement to taking nonrequired mathematics courses in high school. Indicators of SES (parents' education and parents' occupation) were only weakly related to nonrequired mathematics course taking, when other important variables, such as educational aspirations and parental encouragement, were entered in the same regression analysis. Findings from another study indicated that well-educated parents were far more inclined to pressure their children to take demanding mathematics courses (Useem, 1991). Altogether, these findings illustrate an indirect path from SES to advanced course taking through parental expectations and involvement, which influence the student’s own educational aspirations. Peers’ educational aspirations may also play a role in the indirect effect of SES on advanced course taking. Previous research suggests that parents want and encourage their children to have friends with similar educational aspirations, since peers’ aspirations will be influential on the student’s own aspirations (Cooper & Cooper, 1992). Davies and Kandel (1981), in a study of adolescents, found a significant relationship between parents' educational aspirations for their children and aspirations of their children's best friends. Another finding by Lantz and Smith (1981) was that perceived peer attitudes were significantly related to election of nonrequired mathematics courses, even when it was entered in the regression analysis together with parents' education, parents' occupation, parental encouragement, and some other variables.

Direct and Indirect Effects

This group of findings assigns importance to peers’ educational aspirations. An important issue is the difference between parental influence and peer influence on the student’s educational aspirations, which will have an impact on course taking decisions. As for the parent or peer influence in general, peers may have a greater influence in some areas of the student's life, such as types of behavior determining the current adolescent lifestyle, whereas parents may have a higher influence in other issues, such as those relevant to future goals (Biddle, Bank, & Marlin, 1980; Davies & Kandel, 1981). However, with regards to the parent or peer influence specifically on the student's educational aspirations, Davies and Kandel found that parental influence was much stronger than peer influence and did not decline over the adolescent years. Guidance and counseling policies and practices in schools can either magnify or reduce the effect of SES on course taking. There has been an ongoing concern about the availability of counseling services to students and parents, mostly due to high student-to-counselor ratios and counselors’ record keeping duties (Martin, 2002; Powell, Farrar, & Cohen, 1985). A study on a nationally representative sample of public school students in grades five through eleven revealed that a considerable number of students were not told about the academic implications of their course taking decisions in mathematics. Parents did not have any access to counseling services either (Leitman, Binns, & Unni, 1995). In the absence of proper counseling in school, well-educated parents are still able to guide their children through important course taking decisions, since they are knowledgeable about courses. On the other hand, those with low levels of education cannot be of any help to their children in this matter (Useem, 1991). What makes this problem worse is even less availability of these services to low-SES or minority students, who cannot get sufficient—if any— guidance from their families or communities and are in need of guidance the most. Many of these students are concentrated in high-minority, high-poverty schools (Lippman et al., 1996) and are less likely to have access to guidance counseling for course taking decisions (Lee & Ekstrom, 1987; Leitman et al.). The mechanisms mentioned above are ways in which parents’ socioeconomic characteristics can legitimately play an indirect role in students’ course taking. However, there are also concerns sounded by researchers that imply a direct effect of SES, which can be seen as a form of discrimination based on SES. For example, some researchers assert that there is a Mehmet K. Ozturk & Kusum Singh

differential treatment of minority students by school counselors and teachers through counseling them into undemanding nonacademic courses and discouraging them from academic ones (Calabrese, 1989; Leitman et al., 1995). Still others believe that the extent to which schools allow parental involvement in course placements is another factor that works against lowSES or minority students. In schools that try to constrain parental intervention in course taking decisions, parents with high SES or members of the dominant culture have more of the social, intellectual, and cultural resources to acquire the crucial information about course placements and the courage to take initiative. On the other hand, parents with low levels of education remain uninformed and discouraged by the school personnel to take initiative in this matter (Lareau, 1987; Useem, 1991). Previous Mathematics Achievement and Advanced Mathematics Course Taking Findings about the direct effect of previous mathematics achievement on subsequent mathematics course taking are inconclusive. Some studies found that mathematics achievement still had a significant effect on subsequent mathematics course taking, even after taking into account the effect of mathematics selfconcept (e.g., Marsh, 1989); whereas others found that previous mathematics achievement did not consistently predict subsequent mathematics course taking. For example, Lantz and Smith (1981) found that when the last grade earned in mathematics was entered into regression analysis with several other variables, such as parents' education, parents' occupation and the student’s mathematics self-concept, it predicted the participation in nonrequired mathematics courses in only one sample out of three. Students' subjective comparisons of their mathematics performances with those of other students as well as with their own performances in other subjects were better predictors of mathematics participation than the last grade earned. The direct effect of previous mathematics achievement on subsequent mathematics course taking, especially at an advanced level, may be expected, since previous achievement, measured either by grades or standardized test scores, is a widespread criterion in high school mathematics course placements (Oakes, Gamoran, & Page, 1992; Useem, 1991). However, note that its use as a criterion does not necessitate successful students’ enrollment in subsequent nonrequired advanced level mathematics courses.

27

Research Questions The present study tries to answer two important questions: First, is there any discrimination in advanced mathematics course placements based on students’ socioeconomic status? Second, do students who succeed in previous mathematics courses automatically enroll in the more advanced ones? Put differently, is previous mathematics achievement a necessary and sufficient condition for taking advanced mathematics courses; and to what degree do some variables, such as mathematics self-concept and educational aspirations, mediate the relationship between previous mathematics achievement and advanced mathematics course taking? Method Study Design The present study focused on the effects of two background variables—namely, SES and previous mathematics achievement—on advanced mathematics course taking. The purpose of the study is to see how much of the effect of each of these two variables is direct and how much is an indirect effect mediated through parental involvement, educational aspirations of peers, educational aspirations of the student, or mathematics self-concept. This has been done through testing a causal model that includes these secondary variables. The hypothesized relationships are illustrated in Figure 1. SES

School-level variables reflecting school policies that are influential on students’ course taking were controlled by keeping them constant. These variables are type of school, variety and level of mathematics courses offered, graduation requirements, and type of tracking. The following is a description of how and why these variables were kept constant. In this study, only students in public high schools were selected. Two major differences between public and private high schools influenced our decision to ask the research questions stated above for public high school students only. The first difference is in the curriculum. Most private high schools are characterized by a narrow academic curriculum, where all students complete a narrow set of mostly academic courses, almost all of which are required for graduation. Catholic schools, a large sector of private schooling, are a good example of this type of curriculum (Lee et al., 1998; Lee, Croninger, & Smith, 1997). In contrast, public high schools predominantly feature a differentiated curriculum approach (Lee et al.). In these schools, students take a subset of courses among a variety of available offerings and are free to take elective courses beyond graduation requirements. Both research questions stated above are related to students’ course selections and placements. In view of the above-mentioned policies and practices in most private schools, these research questions are primarily relevant to public high school students. EDUC. ASPR.

PARENT INVOLVE MENT

MATH COURSE TAKING

EDUC. ASPR. OF PEERS

PREV. MATH ACH.

MATH SELF CONCEPT

Figure 1. The Causal Model in the Present Study 28

Direct and Indirect Effects

In the present study, course taking in Algebra II, Geometry, Trigonometry, and Calculus is investigated. Accordingly, only students in schools that offered all of these courses were selected.1 School graduation requirements have at least some impact on students’ course taking, since these requirements set the minimum for course taking. Since requiring a minimum of exactly two years of mathematics for graduation was the most common practice in the schools that participated in the NELS: 88 study, only students in such schools were selected in the present study.2 Broadly, tracking can be defined as a way of grouping students, in which students enroll in different programs of study and take different sequences of courses based on their ability and success levels. The multitude of tracking practices in schools makes it difficult for researchers to employ a precise measure of this variable. In much survey research, the three-track categorization (i.e., academic, general, and vocational) has been employed as a crude measure of tracking (Oakes et al., 1992). Literature on the effect of tracking on students’ course taking is confusing and contradictory, since the term ‘tracking’ does not have a uniform meaning. In the present study, only students who were in any one of the three traditional programs (general, academic, or vocational) were selected, which excluded the ones who were in special or innovative programs. Data Source and Sample In this study, the data were drawn from the base year and first and second follow-up of NELS: 88. In the base year, a two-stage stratified sample design was used, with schools as the first-stage unit and students within schools as the second-stage unit (Ingels et al., 1994). The schools were stratified based on type (public versus private), geographic region, urbanicity, and percent of minority enrollment (Spencer et al., 1990). Within each stratum, schools were selected with probabilities in proportion to their estimated eighth grade enrollment, which led to a pool of approximately 1000 schools. In the second stage of sampling, an average of 23 students was selected randomly from each school, producing a total sample of approximately 23,000 eighth-graders for the base year. The first and second follow-up data were collected from the same cohort in 1990 and in 1992, when most of the students were tenth and twelfth graders, respectively. The cohort was freshened with proper statistical techniques in 1990 and in 1992 to achieve a representative sample of the nation’s sophomores in the first follow-up and Mehmet K. Ozturk & Kusum Singh

seniors in the second follow-up, respectively (Ingels et al.). At the first step of sample selection for the present study, students who were members of the NELS: 88 sample in all three waves of data collection (base year, first follow-up, and second follow-up) and for whom transcript data were available, were selected. This group was readily defined by NELS: 88 as a subgroup within the overall sample, and a sampling weight was provided. This group is a nationally representative sample of 1988 8th graders, regardless of whether they graduated from high school four years later or not. Among these students, only those who graduated from high school in Spring 1992 were selected in the first step of this study. At the second step, relevant school variables were kept constant by selecting students who met all of the criteria below: • Did not change their schools between base year (8th grade) and second follow-up (12th grade); • Enrolled in a public school; • Enrolled in schools that offered the complete set of mathematics courses of interest in the present study (Algebra II, Geometry, Trigonometry, and Calculus); • Were in schools that required exactly two years of mathematics for graduation; and • Were either in general, academic, or vocational programs. All the analyses in this study were performed on this sample after a listwise deletion of missing data, which resulted in a sample size of 1,699. Measures Advanced mathematics course taking. This variable was measured as the sum of total Carnegie units earned in Algebra II, Geometry, Trigonometry, and Calculus. A Carnegie unit is defined as “a standard of measurement used for secondary education that represents the completion of a course that meets one period per day for one year” (Ingels et al., 1994, p. O1). Socioeconomic status of the student. SES was measured by F2SES1, a continuous composite variable already available in the NELS:88 database. It was constructed from the base year parent questionnaire data using five items: father’s education level, mother’s education level, father’s occupation, mother’s occupation, and family income. Occupational data were recoded using Duncan’s Socioeconomic Index (SEI) (as cited in Ingels et al., 1994), which assigns values to various occupational groups. 29

Studentâ€™s previous mathematics achievement. This variable was measured by BY2XMSTD, a score from a standardized mathematics test administered in the spring of 1988 (spring of 8th grade for the sample). Parental expectations. This variable was measured by two first follow-up (10th grade) and two second follow-up (12th grade) questions, asking the students how far in school they think their father or mother (one question for father and one for mother) wants them to go. There are ten choices indicating different levels of education. Parental involvement. This variable was measured by two questions asked both in the first follow-up and the second. The first question asked how often the student discussed selecting courses at school with either or both parents or guardians in the first half of the school year. The second question asked how often the student discussed going to college with either or both parents or guardians in the first half of the school year. Choices were never, sometimes, and often. Educational aspirations of peers. This variable was measured by two first follow-up and two second follow-up questions. One question asked the student how important, within the studentâ€™s peer group, it is to get good grades. The second question asked, in the same context, how important it is to continue education past high school. Choices were not important, somewhat important, and very important. Mathematics self-concept. This variable was measured by four first follow-up questions, asking the student to choose the best answer for the following items: 1. Mathematics is one of my best subjects. 2. I have always done well in mathematics. 3. I get good marks in mathematics. 4. I do badly in tests of mathematics. The available choices were false, mostly false, more false than true, more true than false, mostly true, and true. The fourth item was reverse coded for consistency. These four questions in the NELS: 88 database come from the SDQ-II by Marsh (as cited in Ingels, Scott, Lindmark, Frankel, & Myers, 1992). Educational aspirations of the student. This variable was measured by one base-year, one first follow-up, and one second follow-up question asking the students how far in school they think they will get. There were six educational levels as choices for the base-year question, nine for the first follow-up, ten for the second follow-up. 30

Analytic Method Structural equation modeling was used as the analytic technique. Such modeling allowed studying the relative importance of variables as well as their direct and indirect effects on the outcome variable. Recall the hypothesized relationships among the variables in the model (Figure 1). All of the relationships in the model are hypothesized to be positive, meaning that an increase in one variable in a hypothesized relationship leads to an increase in the other. In all analyses, data were weighted by F2TRP1WT, the sampling weight in the NELS: 88 database, created specifically for the sample used in this study. This subsample is described as the students who were sample members in the base year, first follow-up, and the second follow-up of data collection and for whom high school transcripts were collected. An alpha level of .05 was used for all statistical tests. For creation of correlation matrices, standard deviations, and means to be used in structural equation model estimations, the computer program SPSS 10.1 (SPSS Inc., 2000) was used. For structural equation model estimations, LISREL 8.54 computer software (Joreskog & Sorbom, 2003) was employed. Results During confirmatory factor analysis of the measurement model, parental expectations were included as a separate latent variable. However, estimation of this model yielded a high collinearity between parental expectations and educational aspirations of the student. As verification, factor scores for the two constructs were created through exploratory factor analysis, and the bivariate correlation between the factor scores was calculated. The resulting correlation was .798, again indicating a high collinearity. As a result, parental expectations were eliminated from the model, and educational aspirations of the student were kept. Estimation of this revised model yielded a good fit (CFI = .94; GFI = .92; Standardized RMR = .05). With acceptable values of fit indices and all loadings significant at p < .05 level, this model was chosen to be the final measurement model. Table 1 presents the factor loadings. After deciding on the measurement model, the causal relationships among the variables were specified in an initial structural model. Overall, results indicated a good fit with the hypothesized relationships (CFI = .94; GFI = .92; Standardized RMR = .06). The directions and magnitudes of path coefficients, representing the hypothesized relationships between Direct and Indirect Effects

Table 1 Factor Loadings for the Final Measurement Model No. of Math Courses Taken TOTALMAT

Ed. Aspirations of the Student

Math SelfConcept

Ed. Aspirations of Peers

Parental Involvemen t

Previous Math Ach.

SES

1.00

BYS45

.66

F1S49

.79

F2S43

.76

F1S63D

.90

F1S63J

.88

F1S63Q

.90

F1S63S

.69

F1S70D

.65

F1S70I

.72

F2S68D

.43

F2S68H

.47

F1S105A

.48

F1S105G

.58

F2S99A

.50

F2S99F

.64

BY2XMSTD

1.00

F2SES1

1.00

Completely Standardized Factor Loadings. All loadings were significant at p < .05 level.

the variables, were in accord with theory and previous research findings. Therefore, this model was decided to be the final structural model. Total, direct, and indirect effects for the final structural model are given in Table 2. The correlation between SES and previous mathematics achievement was .40. Discussion Findings During the development of the final measurement model, a very high positive correlation was found between parents' educational expectations for their children and students' educational aspirations. This finding supports the claim that, starting from early childhood, children imitate, identify, and, finally, internalize the values and attitudes of their parents (Comer, 1990). It is also congruent with previous findings (e.g., Davies & Kandel, 1981) that parental influence does not decline over the adolescent years. However, the finding in the present study is about the parental influence in matters of future educational Mehmet K. Ozturk & Kusum Singh

plans and should not be overgeneralized to all aspects of adolescent life. One of the two key questions in this study was whether SES would still have a significant direct effect on mathematics course taking after its indirect effects were taken into account. This analysis found no direct effect of SES on mathematics course taking; however, its indirect effect was not trivial (.14). This finding fails to support the claim that parents' SES plays a direct role in students' course placements. It implies that there is no automatic privilege of being a student from a middle- or high-SES family; rather, parental involvement is critical in students taking advanced mathematics courses. When the total indirect effect of SES is partitioned into its components, .09 belongs to the indirect path from SES to educational aspirations of the student to mathematics course taking and .05 belongs to the indirect path from SES to parental involvement to educational aspirations of the student to mathematics course taking. The significant relationship between parental involvement and educational 31

Table 2 Total, Direct, and Indirect Effects for the Final Structural Model

Parental Involvement

Previous Math Ach.

SES

Parental Inv.

Educational Asp. of Peers

Math SelfConcept

Educational Asp. of the Student

Total

.13*

.31*

--

--

--

--

Direct

.13*

.31*

--

--

--

--

--

--

--

--

--

--

Total

.06*

.14*

.47*

--

--

--

Direct

--

--

.47*

--

--

--

Indirect

.06*

.14*

--

--

--

--

Total

.40*

--

--

--

--

--

Direct

.40*

--

--

--

--

--

Indirect Educational Aspirations Peers Math Concept

of

Self-

Indirect Educational Asp. of the Student

No. of Math Courses Taken

--

--

--

--

--

--

Total

.40*

.35*

.46*

.18*

--

--

Direct

.34*

.21*

.38*

.18*

--

--

Indirect

.06*

.14*

.08*

--

--

--

Total

.53*

.14*

.19*

.07*

.17*

.41*

Direct

.30*

.00

--

--

.17*

.41*

.19*

.07*

--

--

Indirect .23* .14* Standardized Total, Direct, and Indirect Effects. * indicates significance at p < .05 level.

aspirations of the student coupled with the finding that there was a very high correlation between parental educational expectations for the student and the student’s own educational aspirations stresses parents’ critical role in their children’s education. Furthermore, our results support the findings of Davies and Kandel (1981) that parental influence on the student’s educational aspirations was much stronger than that of peers and did not decline over the adolescent years. The other key question in this study was to what degree the relationship between previous mathematics achievement and advanced mathematics course taking was mediated by other variables. The direct effect of previous mathematics achievement on the number of mathematics courses taken was found to be .30, which was significant. A direct effect of previous achievement was expected in this study, above and beyond its indirect effect. The finding that previous achievement also had a significant indirect effect of .23 implies that, even though previous achievement may be a necessary condition most of the time, it is not a sufficient condition for students to take advanced and more challenging mathematics courses. When the indirect effect was partitioned, a major portion belonged to the path from previous mathematics 32

achievement to educational aspirations to course taking (.14). The second largest belonged to the path from previous mathematics achievement to mathematics self-concept to course taking (.07). These two components made up almost all of the indirect effect. Since knowledge of mathematics is cumulative and mathematics courses are sequential, these findings suggest that early and continued success in mathematics is critical for maintaining high educational aspirations as well as self-confidence in mathematics. Implications Perhaps the most important finding is that parental involvement plays a critical role in students’ advanced mathematics course taking. Schools can play a major role in improving this determining factor. In the case of advanced mathematics course taking, the first step to enhancing parental involvement is to inform parents about the importance of these courses for the student’s future. This should be done in as many ways as possible, including advising, parent conferences, and sending information to parents. A next step may be to inform parents about the courses, their sequences and prerequisites, and related policies. This, first of all, Direct and Indirect Effects

requires schools being transparent in their course placement policies. Such transparency may also help eliminate claims of implicit discriminatory tracking in many schools. Schools can also clearly state their course placement policies on their websites, in parent manuals, or other related publications. This will encourage parents to get involved in their children’s course taking. Findings from this study also indicate that successful students in mathematics need to translate their achievement into high educational aspirations to continue taking non-required advanced mathematics courses. This translation naturally occurs at home for students from families with a high level of education, where examples of opportunities a strong background in mathematics can provide are immediate. This issue, however, is critical for students coming from disadvantaged families and communities with little appreciation for education and little knowledge of the education system. In the absence of a push by school policies towards advanced coursework, the only source of guidance, encouragement, and support for these students will be their teachers and school counselors. Therefore, frequent individual advising should be provided to such students in order to encourage and motivate them to take advanced courses and alert them to prerequisites and other course placement criteria. Recommendations for Future Research First, a useful follow-up study would check any possible differential relationships among the variables in the model for students in schools with different demographic characteristics. For example, parental involvement may be more important for minority students living in high-minority, high-poverty inner cities than it is for Whites living in low-minority, lowpoverty suburbs. The three school demographic variables that need to be considered are poverty concentration, minority concentration, and urbanicity. Second, future research should also consider perceived utility of mathematics as a possible influential variable. Several studies revealed that the perceived utility of mathematics by students in their future career is a significant factor in shaping educational aspirations based on previous mathematics achievement (Lantz & Smith, 1981; Linn & Hyde, 1989; Reyes, 1984). Due to the limitations of the NELS: 88 database, this variable could not be included in the present study. Finally, qualitative studies investigating the nature of relationships between parents and school

Mehmet K. Ozturk & Kusum Singh

administrators may prove useful in improving students’ advanced course taking through parental involvement. References Anderson, K. L. (1980). Educational goals of male and female adolescents. The effects of parental characteristics and attitudes. Youth & Society, 12, 173-188. Biddle, B. J., Bank, B. J., & Marlin, M. M. (1980). Parental and peer influence on adolescents. Social Forces, 58, 1057-1079. Calabrese, R. L. (1989). The discriminatory impact of course scheduling on minorities. Peabody Journal of Education, 66(4), 30-41. Comer, J. P. (1990). Home, school, and academic learning. In J. I. Goodlad & P. Keating (Eds.), Access to knowledge: An agenda for our nation's schools (pp. 23-42). New York: College Entrance Examination Board. Cooper, C. R., & Cooper, R. G., Jr. (1992). Links between adolescents' relationships with their parents and peers: Models, evidence, and mechanisms. In R. D. Parke & G. W. Ladd (Eds.), Family-peer relationships: Modes of linkage (pp. 135-158). Hillsdale, NJ: Lawrence Erlbaum. Council of Chief State School Officers. (2002). Key state education policies on PK-12 education: 2002. Retrieved March 3, 2006, from http://www.ccsso.org/content/pdfs/KeyState2002.pdf Davies, M., & Kandel, D. B. (1981). Parental and peer influences on adolescents' educational plans: Some further evidence. American Journal of Sociology, 87, 363-387. Finn, J. D., Gerber, S. B., & Wang, M. C. (2002). Course offerings, course requirements, and course taking in mathematics. Journal of Curriculum and Supervision, 17, 336-366. Hoffer, T. B., Rasinski, K. A., & Moore, W. (1995). Social background differences in high school mathematics and science coursetaking and achievement (NCES No. 95-206). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Hossler, D., & Stage, F. K. (1992). Family and high school experience influences on the postsecondary educational plans of ninth-grade students. American Educational Research Journal, 29, 425-451. Ingels, S.J., Dowd, K.L., Baldridge, J.D., Stipe, J.L., Bartot, V.H., & Frankel, M.R. (1994). National education longitudinal study of 1988 second follow-up: Student component data file user’s manual (NCES 94-374). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Ingels, S. J., Scott, L.A., Lindmark, J. T., Frankel, M. R., & Myers, S. L. (1992). National education longitudinal study of 1988. first follow-up: Student component data file user’s manual (NCES 92-030). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Jones, L. V. (1985, March-April). Black-White differences in mathematics: Some new research findings. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL. Jones, L. V., Davenport, E. C., Jr., Bryson, A., Bekhuis, T., & Zwick, R. (1986). Mathematics and science test scores as related to courses taken in high school and other factors. Journal of Educational Measurement, 23, 197-208.

33

Joreskog, K., & Sorbom, D. (2003). LISREL 8.54 [Computer software]. Chicago, IL: Scientific Software International.

Reyes, L. H. (1984). Affective variables and mathematics education. Elementary School Journal, 84, 558-581.

Khattri, N., Riley, K. W., & Kane, M. B. (1997). Students at risk in poor, rural areas: A review of the research. Journal of Research in Rural Education, 13(2), 79-100.

Rock, D. A., & Pollack, J. M. (1995). Mathematics course-taking and gains in mathematics achievement. Statistics in brief (NCES 95-714). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.

Lantz, A. E., & Smith, G. P. (1981). Factors influencing the choice of nonrequired mathematics courses. Journal of Educational Psychology, 73, 825-837.

Schmidt, W. H. (1983). High school course-taking: Its relationship to achievement. Curriculum Studies, 15, 311-332.

Lareau, A. (1987). Social class differences in family-school relationships: The importance of cultural capital. Sociology of Education, 60, 73-85.

Seginer, R. (1983). Parents' educational expectations and children's academic achievements: A literature review. Merrill-Palmer Quarterly, 29, 1-23.

Lee, V. E., Burkam, D. T., Chow-Hoy, T., Smerdon, B. A., & Geverdt, D. (1998). High school curriculum structure: Effects on coursetaking and achievement in mathematics for high school graduates (NCES Working Paper No. 98-09). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.

Spencer, B. D., Frankel, M. R., Ingels, S. J., Rasinski, K. A., Tourangeau, R., & Owings, J. A. (1990). National education longitudinal study of 1988: Base year sample design report (NCES 90-463). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement.

Lee, V. E., Croninger, R. G., & Smith, J. B. (1997). Course-taking, equity, and mathematics learning: Testing the constrained curriculum hypothesis in U.S. secondary schools. Educational Evaluation and Policy Analysis, 19, 99-121. Lee, V. E., & Ekstrom, R. B. (1987). Student access to guidance counseling in high school. American Educational Research Journal, 24, 287-310. Leitman, R., Binns, K., & Unni, A. (1995). Uninformed decisions: A survey of children and parents about math and science. NACME Research Letter, 5(1), 1-10. Linn, M. C., & Hyde, J. S. (1989). Gender, mathematics, and science. Educational Researcher, 18(8), 17-19, 22-27. Lippman, L., Burns, S., McArthur, E., Burton, R., Smith, T. M., & Kaufman, P. (1996). Urban schools: The challenge of location and poverty (NCES 96-184). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Marsh, H. W. (1989). Sex differences in the development of verbal and mathematics constructs: The High School and Beyond Study. American Educational Research Journal, 26, 191-225. Martin, P. J. (2002). Transforming school counseling: A national perspective. Theory into Practice, 41, 148-153. Mayer, S. E., & Jencks, C. (1989). Growing up in poor neighborhoods: How much does it matter? Science, 243, 14411445. Muller, C., & Kerbow, D. (1993). Parent involvement in the home, school, and community. In B. Schneider & J. S. Coleman (Eds.), Parents, their children, and schools (pp. 13-42). Boulder, CO: Westview Press. Oakes, J. (1990). Opportunities, achievement, and choice: Women and minority students in science and mathematics. Review of Research in Education, 16, 153-222. Oakes, J., Gamoran, A., & Page, R. N. (1992). Curriculum differentiation: Opportunities, outcomes, and meanings. In P. W. Jackson (Ed.), Handbook of research on curriculum (pp. 570-608). New York: Mc Millan. Owings, J., Madigan, T., & Daniel, B. (1998). Who goes to America’s highly ranked “national” universities? (NCES 98095). Washington, DC: U.S. Department of Education, Office of Educational Research and Improvement. Powell, A. G., Farrar, E., & Cohen, D. K. (1985). The shopping mall high school. Boston: Houghton Mifflin.

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SPSS Inc. (2000). SPSS 10.1 [Computer software]. Chicago, IL: Author. U. S. Department of Education. (1997). Mathematics equals opportunity. White paper prepared for U.S. Secretary of Education. Retrieved March 3, 2006, from http://ed.gov/pubs/math/index.html Useem, E. L. (1991). Student selection into course sequences in mathematics: The impact of parental involvement and school policies. Journal of Research on Adolescence, 1, 231-250. White, K. R. (1982). The relation between socioeconomic status and academic achievement. Psychological Bulletin, 91, 461481. Wilson, P. M., & Wilson, J. R. (1992). Environmental influences on adolescent educational aspirations. Youth & Society, 24, 52-70.

1

Research suggests that in schools where the variety in low-end (basic, general) mathematics courses is limited, students tend to take more advanced courses and the average achievement in the school is higher (Lee et al., 1998). Although selecting students in schools offering the courses counted as advanced in this study provided control over advanced level course offerings, inability to control for the variety in lower level course offerings should be acknowledged as a limitation of the study. Such control would have significantly reduced the sample size due to the variation in these courses among schools. 2

After the collection of data used in this study, there have been changes in the graduation requirements imposed by states, school districts, or individual schools. These ongoing changes have especially gained momentum after the No Child Left Behind Act of 2001. They include introducing graduation (exit) examinations and requiring specific courses to be taken for graduation. However, this study does not investigate the effect of graduation requirements on course taking, and only deals with it for control purposes. We believe that there have not been any significant changes in the variables investigated through the causal model in this study.

Direct and Indirect Effects

The Mathematics Educator 2006, Vol. 16, No. 2, 35–42

Efficacy of College Lecturer and Student Peer Collaborative Assessment of In-Service Mathematics Student Teachers’ Teaching Practice Instruction Lovemore J. Nyaumwe David K. Mtetwa This study investigated the effectiveness of collaboration between college lecturers and student peers in assessing the instructional practice of in-service student teachers (ISTs). The study was inspired by criticisms that college lecturers’ assessments were not producing valid critiques of ISTs’ mathematical and pedagogical competencies to implement strategies they learned in their coursework. Case studies of two pairs of ISTs, one pair at a state high school and the other at a private high school, provided data for this study. During their coursework, ISTs learned new pedagogical skills and upgraded their content knowledge. Findings indicate that lecturer and peer assessment of the same lesson taught by an IST resulted in different but complementary critiques. The lecturer’s critique highlighted both strengths and weaknesses of a lesson while the peer’s critique refrained from pointing out weaknesses of a lesson. An important implication for the findings, in Zimbabwe, is that the deployment of ISTs in pairs for teaching practice may be beneficial to their professional development.

The purpose of this study is to determine the effectiveness of collaborative assessment of in-service mathematics student teachers’ classroom practice by both a lecturer and a peer. This assessment occurred during the full-time teaching practice segment of their program in which student teachers were encouraged to implement constructivist-inspired instructional strategies. An in-service student teacher (IST) is a certified and experienced teacher who enters a collegebased program of professional study for the purpose of improving their professional knowledge and skills using specific mathematical reforms in pedagogy and content. Peers are fellow in-service student teachers working at the same teaching practice school. Student teachers acting as peers attended the Bachelor of Science in Education (BScEd) in-service program during the same period. A lecturer is a university mathematics educator who teaches the ISTs during the Lovemore J. Nyaumwe is a Mathematics Education lecturer at Bindura University of Science Education, Zimbabwe. He taught mathematics at high school before becoming a lecturer. His research interest is teacher professional development in preservice and in-service contexts. David K. Mtetwa is a senior Mathematics Education lecturer at the University of Zimbabwe. He has a passionate interest for teacher professional development. He has made an impact on improving mathematics teacher education in the Southern Africa region through national and regional involvement as an external examiner for several universities, mathematics education consultant, author and plenary speaker at national and regional conferences.

Lovemore J. Nyaumwe & David K. Mtetwa

residential portion of their program. Lecturer-peer collaborative assessment can be viewed as the joint operation of a lecturer and peer in assessing the level to which an IST has developed attitudes, knowledge, and skills to implement constructivist pedagogical strategies in their teaching. In addition, they make suggestions to improve the implementation strategies specific to an assessed lesson. Constructivist theories encourage the use of learner-centered instructional pedagogies because, from a fallibilist perspective, mathematical knowledge is viewed as context-based. Knowledge is believed to originate from observations, experimentation and abstraction using specific senses and, therefore, is fallible, tentative, intuitive, subjective and dynamic (Nyaumwe, 2004). From a constructivist view, to teach mathematics well is to equip learners with a conceptual understanding of the process skills that enables them to individually or collectively develop a repertoire for developing powerful constructions that concur with viable mathematical knowledge (Davis, 1990). Lecturer-peer assessment involves a lecturer and a peer simultaneously assessing the instructional practice of an IST, or a peer alone making the assessment for the purposes of formative evaluation. In the absence of a lecturer, peer assessment is still collaborative because a peer acts as a proxy for the lecturer and reports to a lecturer when they meet. Reciprocal peer assessment of lessons is a two-way collaborative process that helps both the peer assessor and the IST generate ideas about how to improve their teaching practices. Collaborative evaluation of an IST’s implementation of a 35

pedagogical reform is essential in order to identify strengths and weaknesses. If the IST is performing unsatisfactorily, the evaluation can assist in upgrading practice to appropriate standards (Ziv, Verstein, & Tamir, 1993). Assessment of ISTs’ professional competencies during practice teaching is a polemical issue in Zimbabwe. Teacher educators in the country take the assessment as their privileged domain. They lament subjectivity and the propensity toward bias when school authorities get involved in the assessment process. Usually school-based assessments and lecturers’ assessments vary significantly (Nyaumwe & Mavhunga, 2005). Despite the assessment differences, Zindi (1996) suggested that schools be involved in the assessment of student teachers. He argued that reliance on a lecturer as the only assessor of student teachers’ practice was not fair or valid because several assessors produce more objective assessments of student teachers’ professional competencies than a single source. Mathematics curriculum reform in Zimbabwe encourages teachers to adopt constructivist approaches in their teaching because of the potential of these approaches to enable learners to transfer school mathematics to contextualized situations through modeling and problem solving. Constructivist strategies emphasize (a) linking content to learners’ prior knowledge, (b) analyzing and interpreting learners’ thinking and understanding, (c) encouraging learner construction of mathematical concepts and negotiations, as well as (d) facilitating multiple presentations of solutions to problems. Constructivist strategies also involve ISTs’ abilities to experiment with new approaches that require learners’ engagement in well-developed, open-ended, and authentic investigations. In these investigations, learners develop and evaluate conjectures, explain their work, and communicate their results. Due to differences in learner characteristics and difficulties in effective sampling of instructional skills across the domain of constructivist tenets, arguably, the traditional lecturer assessments of ISTs’ professional competencies do not provide a valid measure of the IST’s ability to implement constructivist strategies (Watt, 2005). This observation suggests it may be useful to incorporate peers in the assessment process in order to capture a wider range of instructional abilities viewed from different perspectives. The present study was inspired by an interest to explore the relationship between assessment measures from a variety of assessors for ISTs’ classroom practice during the 36

student teaching segment of their BScEd program. It attempts to contribute to the debate on this issue by investigating the research question: Does lecturer-peer assessment of classroom practice of ISTs enhance implementation of constructivist-related strategies when teaching? Answers to this research question could inform local and international discussion on promoting holistic assessment of ISTs’ instructional practice. Findings from this study could also have important implications for the deployment of ISTs to teaching practice schools in Zimbabwe and elsewhere. Conceptual Framework from Theoretical Considerations When working in cooperative groups, the involvement of peers in the acquisition of procedural and conceptual understanding of mathematical content is well documented (Lowery, 2003; Schmuck and Schmuck, 1997). Watt (2005) suggested that, in Australia, the use of peers to assess each other’s work has potential for improving learner mastery of mathematical knowledge and skills. Viriato, Chevane, and Mutimucuio (2005) explored the degree to which peer assessment could contribute to the acquisition of generic competencies of post-graduate science learners in Mozambique. Both studies concluded that peer assessment fosters deep understanding, increases learner involvement in the academic life, contributes in the development of reflective skills, increases awareness of a broad range of possible solutions to problems, contributes to the development of self– reliant and self-directed learners, and increases cooperation and social interaction by lessening competition among learners. (Sivan, as cited in Viriato et al., p. 23)

These studies depict peer involvement in learning as having a positive effect because it enhances understanding of subject matter content. Morrison, Mcduffie, and Akerson (2005) proposed that, when teachers work with peers, the application of new knowledge in appropriate contexts is facilitated by their negotiations of and active involvement in the implementation of the knowledge, and in watching and discussing the efficacy of the implementation strategies used. This is particularly true for the development of instructional skills that develop during active implementation in real classrooms. Putnam and Borko (2000) argue that, in order to construct new knowledge about pedagogical reform, ISTs need to be situated in authentic classroom contexts. Immersion in these Efficacy of College Lecturer and Student Peer Collaborative Assessment

contexts when implementing a pedagogical reform promotes the transfer of theoretical knowledge of the reform to practice. Lecturer-peer assessment of ISTs’ implementation of instructional strategies encourages ISTs to study how peers interpret and implement pedagogical reform. Well-organized peer assessments might not only focus on peer understanding of instructional theory and practice but also enhance the development of a repertoire of professional skills through explanations, justifications of claims, and communication with peers during post-lesson reflective dialogues. The use of peers in a learning environment has been documented as beneficial to the development of a deep understanding of what peers and ISTs learn collaboratively (Watt, 2005; Viriato et al., 2005). Peers have the potential to expose each other to reform strategies and techniques, share personalized strategies and techniques, and collaborate in the evaluation of implementing a pedagogical reform. Spector (1999) recommends having ISTs sit in peers’ lessons in order to help each other to better understand and apply the theories of a reform and implement them in their teaching. Context of the study In-service education is an individual teacher’s personal initiative in Zimbabwe. There are numerous motivations for embarking on in-service training. Teachers who obtained certificates or diplomas in education from a teachers’ college are qualified to teach middle secondary school mathematics. For those teachers to teach high school mathematics, increase chances for promotion, or get a higher remuneration, they must enroll in a full-time undergraduate in-service program at a state or private university (for more information on the Zimbabwean educational system, see Appendix A). The Ministry of Education, Sport and Culture of Zimbabwe supports that initiative by granting two-year leaves of study to tenured teachers. This study focuses on mathematics ISTs enrolled in a science education program at a state university located in the northern part of Zimbabwe. To graduate with the BScEd degree from this university, an IST has to pass 24 content courses in mathematics and complete a dissertation, a practicum, and four professional courses in education and mathematics pedagogy. The ISTs enroll in six mathematics content courses per semester that are also offered to preservice undergraduate majors. They study all of the mathematics content courses in the undergraduate

Lovemore J. Nyaumwe & David K. Mtetwa

program in order to meet the certification requirements of the BScEd degree offered by the university. The ISTs enroll in one professional course per semester separate from the preservice undergraduate students. The ISTs are exempted from some undergraduate education courses under the assumption that they possess sufficient knowledge and skills acquired during their initial training at a teachers’ college. The program has more content than pedagogy courses because it is assumed that the ISTs already have a pedagogical base and that they need to transform it into a learner-centred orientation so as to facilitate the implementation of constructivist instructional strategies in their teaching. After completing the Advanced Pedagogics course, the ISTs participate in four weeks of teaching practice in between semesters of the program’s second year. The goals of the Advanced Pedagogics course are that inservice students develop (a) a theoretical framework for teaching mathematics at the high school level, (b) a repertoire of constructivist theories for teaching mathematics, (c) favorable attitudes toward mathematics and mathematics teaching, (d) an understanding of the importance of modeling and problem solving in a context accessible to the learner, and (e) the ability to apply the knowledge and skills acquired in the course. ISTs that pass all prerequisite courses for the teaching practicum independently look for and select schools for this experience. Schools accept ISTs after agreeing to the conditions that ISTs teach under a qualified cooperating teacher for four weeks and that they observe classes taught by their cooperating teacher as well as other teachers in the mathematics department. The participants in this study were deployed within a 200 km radius of the university. Method A convenience sample (Watt, 2005) from a cohort of 22 ISTs was used for data collection. The only criterion for sampling was attendance of at least two in-service mathematics student teachers at the same high school. Both high schools that met the sampling criterion were located in an urban setting. In the first visit by a lecturer, a common understanding of lecturer-peer assessment was made by reviewing the previously described characteristics of constructivist-inspired pedagogies in vogue at the university. The lecturer-peer assessment was meant to be formative rather than summative. To achieve this goal, the assessments were made on the basis of each assessor’s personal impression, understanding, and 37

perceptions of constructivist tenets. In addition, the assessors were free to consult the constructivist pointers on the official classroom observation instrument used by the university (see Nyaumwe & Mavhunga, 2005, for the instrument). Assessors were encouraged to use their personal understanding of constructivist instructional strategies since a standard assessment instrument may force them to focus on a uniformly restricted sample of instructional skills. Two lecturer-peer assessments of lessons taught by each of the two peers, making a total of four lessons, were made at each of the participating schools. To increase the reliability of the assessment process, participants conducted multiple assessments of the lessons before data collection. After reaching a common understanding on how lessons could be assessed, the lecturer and peers made independent assessments. A lecturer and a peer sat in the same lesson delivered by an IST and produced individual lesson assessments. Each assessor took detailed notes of classroom episodes. These notes were used in the postlesson reflective dialogues to pose and support assertions made during lesson observation. One copy of the written field notes was given to the IST who taught the lesson in order to facilitate personal reflection. Another copy was given to the lecturer, who was also the researcher, for the purposes of this study. The post-lesson reflective dialogues were audio taped and later transcribed. The assessed lessons were typically 70 minutes long. The researcher interviewed each student teacher pair separately at the end of the four-week school attachment to determine the professional benefits gained from the lecturer-peer assessment. Data were analyzed by interpretative and analytic induction (Bogdan & Biklen in Morrison et al., 2005) by judging the extent to which the instructional practice was commensurate with constructivist instructional strategies. Similarities and differences between the instructional practice and constructivist perspectives were evaluated and recorded on assessment critique forms.

Results from Written Critiques The critiques written by the lecturer and a peer on the performance of an IST’s instructional practice highlighted some similarities and differences. The following written critiques of a lesson taught by James involving arithmetic and geometric series serve as examples of the lecturer-peer assessment data produced during the study. Throughout the critiques below, the IST’s instructional actions are described by the assessor verbatim, whereas the assessor’s interpretations of instructional actions using constructivist tenets are presented in brackets. The lecturer wrote the first critique:

Results

The IST went round the classroom to listen and assess learners working in groups. [It can be assumed that he evaluated learners’ comprehension and how they were solving the problems]. Group work was concluded by highlighting the steps that can be taken when finding the solution for number 4 that read “the third term of a GP is 8 and the fifth term is 32, find S5”. [Instead of the IST using a question and answer session to formulate the simultaneous equations and subsequently finding

1

A written critique from James and Elizabeth , student teachers attached at the government school, and an excerpt of an interview of Beaven and Munashe, student teachers based at the private high school, are used to provide evidence of the efficacy of lecturer-peer assessments.

38

The introduction was on the conceptual meaning of arithmetic and geometric series. The learners were asked to provide the definitions and formulae for the nth terms and sums of an AP and a GP. [Learners’ previous knowledge was determined in the introduction of the lesson]. Learners were later asked to solve problems on work cards in groups of between 3 and 5 each. The learners were encouraged to solve the problems on the work cards using six pre-determined steps of comprehension, identification of variables, the question/task, rule to use, making substitutions and solving the equation. [The idea of using groups was good to enable learners to socially invent their solutions but group sizes were too large for meaningful learner trial and error and negotiations of solution methods. The provision of several tasks on a work card catered for learners’ individual differences by outcome. The pre-determined steps given for learners to use as guide to solve the problems reveal the IST’s formalist conception of mathematics learning where ‘correct’ answers are obtainable from using formal rules, procedures or formulae]. The tasks on the work cards were thought provoking for example, suppose θ+ Φ+ В+ Ω + … is an arithmetic progression with a common difference , find S4. [This and similar problems required more reasoning and understanding than direct application of a formula. One can conclude that they were more of problem-solving than consolidation exercises].

Efficacy of College Lecturer and Student Peer Collaborative Assessment

the values of the first term, a, and common ratio, r, different groups should have presented their solutions on the board. This was going to allow the emergence of possibly different solution strategies that might provide similar viable values for S5].

The assessment made by a peer of James for the same lesson is as follows: There was a good use of learners’ prior knowledge by asking the nth terms and sums of the arithmetic and geometric progressions. [Learners’ current knowledge was linked to new concepts to be developed in the lesson. This enhances building of new concepts on knowledge existing in learners’ memories]. Learners’ interests were aroused by asking them to come to the board to write the formulae for sums and terms to nth terms of arithmetic and geometric progressions. The learners were asked to explain the formulae that they wrote on the board. [Probing of learners’ responses to ascertain their current understanding of the formulae was appropriate. It assisted James to ascertain learners’ current understanding of arithmetic and geometric formulae]. The progression of the lesson was logically arranged from the known - formulae of AP and GP - to the unknown - application of the formulae in novel questions in groups. [James utilized learners’ prior knowledge. This was revealed by the formulae they recalled for solving problems on work cards]. James used various methods to solicit learners’ understanding such as verbal, written work on the board and in groups and listening to group discussions. [James continuously evaluated learners’ understanding from time to time during the lesson]. James encouraged learners to debate their solutions, for instance, when formulating the simultaneous equations on the problem on GP. [Debating and negotiating viable solution methods characterized the norms of James’s instructional practice]. The tasks on work cards were challenging and generally they suited the competence levels of learners. James was confident and knowledgeable of the content under review. [The work covered in the lesson catered for learners’ individual differences because it seemed to suit their cognitive levels. Teacher confidence and knowledge are prerequisite for successful instruction]. Though there were some elements of noise and movement of learners in the lesson, they were justifiable as they facilitated discussions and verifications of learners’ current understanding with peers in different groups. [Learner-centered Lovemore J. Nyaumwe & David K. Mtetwa

lessons are necessarily characterized by disagreements in the initial stages that lead to learner negotiations and finally to a consensus. Some element of noise and movements are permissible as learners consult, verify their conjectures and explain to each other what they think is viable and justifying their decisions]. James interacted with the learners individually, in groups or at class level during presentations on the board. [High teacher-learner interaction is recommended for teachers to be aware of learners’ current thinking, understanding of concepts and using it to develop new concepts].

Although the lecturer and peer’s critiques are based on the same lesson, they highlight different constructivist pedagogical skills evident in an IST’s instruction. For instance, the lecturer did not comment on the IST’s pedagogical and mathematical competencies, but the peer highlighted them as important to determine successful implementation of constructivist instructional strategies. IST mastery of content is critical because a knowledgeable teacher anticipates alternative conceptions and solution strategies by learners. Whereas the lecturer expected learners to present their group work solutions on the board as a way of soliciting multiple methods for finding a solution of a problem, the peer highlighted learners’ movement in the classroom during group work. Both the lecturer and the peer concurred on highlighting differentiation as a constructivist tenet exhibited by James. The peer’s critique specifically refrained from pointing out weaknesses in James’s implementation of constructivist instructional practice. These critiques demonstrate that the peer did not identify weaknesses in the IST’s practice. Peers, in general, were hesitant to identify weaknesses of a lesson because they perceived such criticisms as more summative than formative. In addition, the peer identified constructivist tenets that were not identified by the lecturer and vice versa, providing complementary assessments. Results from Interviews of Student Pairs Both pairs of ISTs hailed lecturer-peer assessments as a major benefit of teaching practice with peers. The personal experiences described by Beaven, one of the ISTs involved, were similar to those echoed by other peers: The collaborative lecturer and peer assessment gave me a great learning experience. The lecturer and the peer looked at different professional competencies in the same lesson. The use of episodes from the lesson during post-lesson 39

reflective dialogues and the constructivist/ absolutist theory that explains it from the understanding of a peer and a lecturer provided me with a wide perspective of how a teaching episode can be a point of focus for one assessor and a trivial event to another. Though my peer was a novice in the area of classroom assessment he provided constructive critiques to the lessons I taught.

The joint assessments enabled the ISTs to gain insight into the interplay of curriculum goals, school contexts, content, learners, the learning milieu and the constructivist/absolutist instructional strategies they learned during teacher education coursework. The lecturer-peer assessments and subsequent post-lesson reflective dialogues exposed the ISTs to new strategies and techniques of implementing constructivist instructional strategies. Discussions in the post-lesson reflective dialogues provided peers with a wide range of interpretations of their instructional practices. The suggestions made during these dialogues enhanced IST understanding of the implementation strategies that promote learner understanding. Based on the interview excerpt and lesson assessment critiques, the lecturer-peer assessment enhanced ISTs implementation of constructivist-related strategies when teaching mathematics during their teaching practice. Discussion Findings from this study indicate that the use of lecturers and peers to assess ISTs’ implementation of instructional strategies is beneficial to the development of their professional skills. While simultaneously assessing a lesson, a lecturer and a peer focus upon and interpret instructional actions differently because their individualized beliefs and values about teaching and learning are filtered through personal frames. For instance, Munashe applauded Beaven for stating the objectives at the beginning of a lesson as a motivational strategy, while the lecturer perceived it as a way of enabling learners to understand the sequence of the lesson. Joint assessment of a lesson involves a lecturer and peer using individualized perceptions of an instructional episode. The complexity of these individualized perceptions make it impossible for any two evaluators assessing the same lesson to see and interpret the professional competencies in the same way, even in the presence of an agreed common instrument. Fortunately, any differences in the assessments highlight complementary teaching skills which, when combined, provide a synergy of an IST’s 40

pedagogical competence to implement desired strategies. Teaching mathematics is a complex interpretive process that depends on the context of the learning environment, nature of content, learner interest, school ethos, and curriculum goals, among others. Based on this complexity, the pedagogical competencies of an IST do not rest on a universally “accepted set of facts, rules and assumptions” (Steele, 2005, p. 295). The pedagogical competencies are not static to allow the use of predetermined indicators as they vary in response to learner needs. Learner behaviors and a teacher’s interpretation of the learning environment makes teaching vary from one moment to another, making replication of a teaching episode impossible (Steele, 2005; Wilson, 2003). Lecturer-peer assessment liberates IST assessments from the personalized beliefs and expectations of lecturers. As it bases assessment on a variety of opinions, helping ISTs to explore a variety of pedagogical ideas, lecturer-peer assessment might, in turn, enhance their implementation of desired pedagogical practices. The use of one source of assessment data on ISTs’ classroom practice is not adequate because teaching mathematics is an interpretive act (Steele, 2005) that depends on an individualized area of focus. Because there are no clearly defined rules for assessing pedagogical competencies, assessment of ISTs’ instructional practice is conducted with the cooperation of different sources of players. Multiple assessors are necessary because teaching is an interpretative act and assessments are conducted using a frame that is contextualized and individualized. Multiple sources of assessment data on the instructional practice of ISTs might facilitate an understanding and development of instructional knowledge and skills (Peressini, Borko, Romagnano, Knuth, & Willis, 2004). Involvement of peers in assessment has motivational and cognitive merits. From a motivational perspective, peer collaborative assessments contribute to feelings of control regarding how the ISTs learn, gain confidence, and understand how to implement constructivist instructional strategies in their teaching. In the post-lesson reflective dialogue, a peer and a lecturer identify an episode from the assessed lesson and use personalized understanding of constructivist tenets to interpret it. The interpretations and explanations of an instructional episode lead to a discourse. Discourse on pedagogy provides opportunities for peers and the lecturer to reflect, make and defend claims, exchange alternative perceptions,

Efficacy of College Lecturer and Student Peer Collaborative Assessment

and negotiate a consensus that can be generalized to instructional practices of other concepts. For instance, in a problem-solving task, Elizabeth asked learners to share 17 cattle in three groups, consisting of ½, ⅓ and 1/9 of the cattle, respectively. The learners shared the cattle and obtained 8.5, 5.7 and 1.9 cattle, which they rounded off to 9, 6 and 2. In the post-lesson reflective dialogue the lecturer and James questioned the reasonability of a non-integer quantity of cattle. They argued that the approach caused the mathematical results to become abstract and meaningless. A logical way of approaching the ratio problem that maintains realistic mathematical results was to make a total of 18 cattle by borrowing one cow and adding it to the 17 that were available. Eighteen cattle can be shared evenly using the fractions ½, ⅓ and 1/9 and the borrowed cow can be returned after sharing. Further discussions on the problem revealed that, by rounding off results, learners could obtain viable answers from flawed reasoning. Logical reasoning is an important skill in the mathematics curriculum that learners should be given opportunities to develop. Arguments, like the one in the post-lesson reflective dialogue, have potential to deepen ISTs’ practical and theoretical understanding of constructivist instructional strategies in ways that might enhance learner achievement. For the purposes of formative evaluation, the use of peers to complement lecturers’ assessments of ISTs is a viable initiative for producing a holistic picture of their classroom practice. Lecturers expose the pedagogical strategies that they wish ISTs to implement in the classroom, making them pursue similar strategies. In contrast, school authorities may need training in order to understand these strategies that ISTs are required to adopt. A lack of coherence between the university and schools enables school authorities to emphasize teaching skills that are in conflict with those encouraged by lecturers (Nyaumwe, 2001; Nyaumwe, Mtetwa & Brown, 2005). Reliance on lecturer assessments of teaching practice has been justified on the grounds that lecturers were perceived as impartial and that their assessments would maximize reliability and ensure comparability of ISTs’ attained instructional competencies. This belief devotes most of the “investment in assessment on certification and accountability to the neglect of formative evaluation” (Black, 1998, p. 812). An influential reason for Zimbabwean teacher educators’ resistance or indifference to allowing other forms of school-based assessment is that they regard them as highly subjective (Nyaumwe & Mavhunga, 2005). Lovemore J. Nyaumwe & David K. Mtetwa

The peer’s assessment in this study did not critique the ISTs’ lesson delivery because it was perceived as summative rather than formative evaluation. Peer assessments may not be valid for summative evaluation where assessments are used to rank students according to ability, certification or accountability purposes. This study did not attempt to assess peers’ ability to evaluate each other’s instructional practices because it was concerned with formative evaluation only. The extent to which peers’ evaluations are valid might form the focus of another study. The view of lecturers as the sole assessors and evaluators of ISTs’ classroom practice in Zimbabwe is held at the expense of validity because lecturers’ assessments may mask some weaknesses in implementing desired instructional practices. Findings from this study have shown that the lecturer-peer assessments are effective for the purpose of formative evaluation of ISTs’ instructional practice. Debates among Zimbabwean educators on whether or not to adopt this model of assessment might be informed by their preference to prioritize formative or summative evaluation. One case study cannot amplify all the merits and demerits of the lecturer-peer assessment phenomenon and the variables that might influence its success. A similar study on preservice student teachers might be useful in order to begin to see the phenomenon in a wider frame. References Black, P. (1998). Assessment by teachers and the improvement of students’ learning. In B. J. Fraser & K. G. Tobin (Eds.), International handbook of science education (pp. 811–822). Dordrecht, Netherlands: Kluwer Academic. Davis, R. B. (1990). Constructivist views on the teaching and learning of mathematics. In R. B. Davis., C. A. Mahler, and N. Noddings (Eds.), Constructivist views on the teaching and learning of mathematics. Reston, VA: National Council of Teachers of mathematics. Lowery, N. V. (2003). Assessment insights from the classroom. The Mathematics Educator, 13(1), 15–21. Morrison, J. A., Mcduffie, A. R., & Akerson, V. L. (2005). Preservice teachers’ development and implementation of science performance assessment tasks. International Journal of Science and Mathematics Education, 3, 379–406. Nyaumwe, L. (2001). A survey of Bindura University student teachers' perceptions of the mentoring model of teaching practice. Zimbabwe Journal of Educational Research, 13(3), 230–257. Nyaumwe, L. (2004). The impact of full time student teaching on preservice teachers' conceptions of mathematics teaching and learning. Mathematics Teacher Education and Development, 6, 23–36.

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Nyaumwe, L. J., & Mavhunga, F. Z. (2005). Why do mentors and lecturers assess mathematics and science student teachers on school attachment differently? African Journal of Research in Mathematics, Science, and Technology Education, 9(2), 135– 146. Nyaumwe, L. J., Mtetwa, D. K., & Brown, J. C. (2005). Bridging the theory-practice gap of mathematics and science preservice teachers using collegial, peer and mentor coaching. International Journal for Mathematics Teaching and Learning. Retrieved July 14, 2005, from http://www.ex.ac.uk/cimt/Ijmtl/nyaumwe.pdf Peressini, D., Borko, H., Romagnano, L., Knuth, E., & Willis, C. (2004). A conceptual framework for learning to teach secondary mathematics: A situative perspective. Educational Studies in Mathematics, 56, 67–96. Putnam, R., & Borko, H. (2000). What do new views of knowledge and thinking have to say about research on teacher learning? Educational Researcher, 29, 4–15. Schmuck, R. A., & Schmuck, P. A. (1997). Group processes in the classroom. Boston: McGraw Hill. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4–14.

Steele, M. D. (2005). Comparing knowledge bases and reasoning structures in discussions of mathematics and pedagogy. Journal of Mathematics Teacher Education, 8, 291–328. Viriato, N., Chevane, V., & Mutimucuio, I. V. (2005). Student peer assessment in a science education competence-based course. In C. K. Kasanda, L. Muhammed, S. Akpo, & E. Nyongolo (Eds.), Proceedings of the 13th Annual SAARMSTE Conference (pp. 119–125). Windhoek, Namibia: Namibia University. Watt, H. (2005). Attitudes to the use of alternative assessment methods in mathematics: A study with secondary mathematics teachers in Sydney, Australia. Educational Studies in Mathematics, 58, 21–44. Wilson, S. M. (2003). California dreaming: Reforming mathematics education. New Haven, CT: Yale University Press. Zindi, F. (1996). Towards the improvement of practical teaching assessment. The Zimbabwe Bulletin of Teacher Education, 4(4), 26–37. Ziv, S., Verstein, M. S., & Tamir, P. (1993). Discrepant evaluations of student teacher performances. Education Research and Perspectives, 20(2), 15–23.

Spector, B. S. (1999, March). Bridging the gap between preservice and inservice science and mathematics teacher education. Paper presented at the annual meeting of the National Association for Research in Science Teaching, Boston, MA. 1

Pseudonyms are used for moral and ethical reasons to protect the identities of the participants.

Appendix A Table A1 Levels of Formal Education in Zimbabwe Student Age

Education Level

Required Teaching Credential

3–5

Kindergarten

6–12

Primary

13–14

Secondary: ZJC

15–16

Secondary: ‘O’ Level

Diploma: Teachers’ College

17–18

Secondary: ‘A’ Level

Diploma: Teachers’ College

19–22+

Tertiary (Undergraduate)

Degree: University

Note. ZJC = Zimbabwe Junior Certificate, ‘O’ Level = Ordinary Level, ‘A’ Level = Advanced Level. Candidates who pass ‘O’ Level but fail to enroll in ‘A’ Level or who pass ‘A’ Level but fail to enroll at a university may study for a diploma in teaching at a teachers’ college. Those who pass ‘A’ Level and enroll at a university may study for an undergraduate degree program..

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Efficacy of College Lecturer and Student Peer Collaborative Assessment

The Mathematics Educator 2006, Vol. 16, No. 2, 43–46

In Focus… Mentor Teachers’ Perspectives on Student Teaching Ginger A. Rhodes, Jeanette Phillips, Janet Tomlinson & Martha Reems As a beginning university supervisor, Ginger had limited experience and thoughts about student teaching. After three years of observing student teachers, she’s come to realize that university supervisors and mentor teachers have a range of ideas about the purpose of student teaching, as evidenced by their interactions with and requirements of student teachers. In the past two years, she has observed student teachers at one school where the mentor teachers, Janet, Jeanette, and Martha, have made a conscious effort to reflect upon their mentoring strategies. The mentor teachers participated in the project Partnerships in Reform in Mathematics Education (PRIME), a component of the NSF-funded Center for Proficiency in Teaching Mathematics (CPTM).1 PRIME is a multi-level professional development program for preservice teachers, mentor teachers, and university supervisors at the University of Georgia. Each of the three groups of educators— preservice teachers, mentor teachers, and university supervisors—investigated their own practice while participating in this two-semester professional development program. As a part of PRIME, the three groups of educators met on a weekly basis for 30-60 minutes. This paper presents a collection of the ideas and beliefs of these mentor teachers who go beyond providing the typical student teaching experience. In a discussion Ginger had with the three mentor teachers, two major ideas regarding the student teaching experience emerged along with three questions for the education community. Through this paper we hope to stimulate discussion about the purpose of student teaching and the role of the mentor teacher.

Ginger Rhodes is a doctoral student in Department of Mathematics and Science Education at the University of Georgia. Her research interests include teachers’ understanding and use of students’ mathematical thinking in teaching practices, teacher education, and secondary students’ geometric reasoning. Jeanette Phillips, Janet Tomlinson, and Martha Reems are Nationally Board Certified high school mathematics teachers in Georgia.

Ginger A. Rhodes, Jeanette Phillips, Janet Tomlinson & Martha Reems

Two Important Ideas First and foremost, student teachers should work in a nurturing and supportive environment that encourages them to experiment. Some have used the phrase “sink or swim” to describe the situation student teachers and beginning teachers sometimes face when learning how to teach. This idea is grounded in the belief that one learns best how to teach through experience. We believe this perspective is detrimental to the student teaching experience. We agree experience provides enormous opportunities for growth and development. Yet, giving full responsibility too early to beginning teachers will influence what they learn. For example, suppose a student teacher wants to incorporate into a lesson a paper folding activity that entails managing small groups and whole group discussions. If the student teacher has not developed the skills to observe students’ mathematical thinking during this activity, then it is asking too much of that student teacher to also manage supplies and monitor student behavior. Many times, mentor teachers and university supervisors believe student teachers need to manage an entire lesson on their own, a seemingly obvious requirement for those aspiring to be a full-time teacher. Indeed, many preparation programs require student teachers to carry a full-load of classes for a minimum of two weeks, but there is typically some flexibility. It is our belief that student teachers should not be given this full responsibility too quickly or left on their own to figure out what teaching entails. In high schools, the timeline for acquiring new classes should be determined based on the readiness of the student teacher. When student teachers are learning to create and implement lessons, it is acceptable to divide responsibilities for the lesson between the mentor teacher and student teacher. This provides an opportunity for the student teacher to focus on a piece of the lesson instead of being overwhelmed with all aspects. For example, the student teacher could facilitate the tasks in a lesson while the mentor teacher walks around to manage behavior and classroom organization (e.g., passing out supplies and equipment, 43

collecting papers, etc). This allows and encourages the student teacher to incorporate various teaching ideas (e.g., a hands-on activity, use of technology, and small group work) without the pressures of managing the entire lesson. It is important for student teachers to develop strategies for managing behavior, but it is also valuable for student teachers to notice their students’ mathematical thinking during activities. By removing most behavior management issues during beginning lessons, the student teacher is allowed to focus attention on instructional decisions. As a student teacher gains more experience with lessons and the ability to develop and implement portions of lessons, he or she should take on more responsibility, including managing student behavior. A criticism against mentor teachers taking an active role with student teachers in lessons is that K-12 students may lose respect for the student teacher. We believe this can be prevented in several ways. On their first day in the classroom, student teachers should be introduced to students as a co-teacher, or as another teacher in the classroom. From that point on, they can interact with students. For example, student teachers can help students during activities, go over homework, or work with small groups. We want to emphasize that while they are not taking on large roles, they are, more importantly, interacting with students. In addition to student teachers immediately taking an active role, the mentor teachers should be conscious of the ways they interject comments while their student teacher is interacting with students. In other words, if the mentor teacher treats the student teacher as a colleague in front of the students, the students will view the student teacher as a teacher. This leads to our second major idea. Secondly, student teachers should be treated as colleagues. This may seem obvious to some, as student teachers have reached the end of their coursework and the beginning of their teaching careers. To some, this may seem surprising, as student teachers are inexperienced in the classroom and lack the knowledge gained from experience. We believe that treating student teachers as colleagues provides them with insight into the teaching profession that will support their development as reflective practitioners. Mentor teachers will also benefit, as these conversations will most likely support their own professional growth. As teachers, we make instructional decisions in the classroom from moment to moment. To an observer, these decisions (and their underlying rationale) may not be obvious. It is important for mentor teachers to make these decisions explicit in order for student 44

teachers to gain awareness of issues dealing with mathematics, students, and pedagogy. Thus, mentor teachers should engage student teachers as colleagues in professional conversations. These conversations can happen through co-planning and co-teaching activities, analyzing student work, and attending and discussing professional meetings. As educators, we believe that telling is a less effective way of teaching content to students. If we assume students learn mathematics best when they are engaged in mathematical thought, then it is only natural to assume that student teachers should learn about teaching mathematics in a similar manner. Student teachers have a thirst for knowledge about teaching mathematics. They are excited to learn and want to be successful teachers. Often, they will search for the “correct” way to teach and look to other educators for answers. It is easy for mentor teachers to fall into the trap of telling the student teacher to teach this content in this particular way. Student teachers benefit more from being involved in professional conversations about planning, implementation, and reflection. For example, when Janet discusses planning lessons with her student teacher, she tends to ask for his input. Instead of the student teacher implementing lessons that Janet or he pre-planned individually, the student teacher is implementing lessons that they jointly planned. In order to highlight what we mean be treating student teachers as colleagues, we wish to share three additional examples from our experiences with student teachers. Last year, Janet and her student teacher regularly graded tests and quizzes together. During this time, Janet would verbalize her reflections on how her students were thinking mathematically, question particular solutions, question particular test items, and connect students’ performance to classroom practices. Through these conversations, her student teacher developed similar habits of thought. Another example we wish to share is Jeanette inviting and encouraging her student teacher to attend meetings and conferences with her. In particular, Jeanette and her student teacher regularly attended county meetings that focused on teaching AP Calculus. These meetings consisted of a group of teachers who were teaching AP Calculus and who came together for the purpose to discuss and improve their teaching practices. After these meetings, Jeanette and her student teacher had conversations in order to share and elicit thoughts. Not only did the student teacher observe and participate in the meetings, she had

Mentor Teachers

professional conversations about the meetings with Jeanette. The final example we wish to share is Martha’s student teacher videotaping a lesson on synthetic division. Then, during our weekly PRIME meeting we watched a portion of the video. The discussion of the video began with the student teacher sharing her thoughts and questions about the lesson. In the conversation that followed, the mentor teachers did not simply answer the student teacher’s questions or tell her how they typically teach synthetic division. Rather, the conversation focused on reflection. The group discussed the purpose of the lesson and the ways high school students engaged the mathematics found in the lesson. The group then discussed the mathematics behind synthetic division, which led to other ways to introduce it to students. We believe this kind of mathematical and pedagogical conversation can and should exist between mentor teachers, student teachers, and university supervisors. The two ideas we presented in this section can be viewed as contrasting ideas. One may question how a mentor teacher can treat a student teacher as a colleague and protect him from becoming overwhelmed by the complexities of teaching. We don’t believe a mentor teacher has to do one or the other; she can support and nurture a colleague. Teaching mathematics is complex and difficult at times. Student teachers need to have a realistic view of what teaching entails. We are suggesting that mentor teachers, as well as university supervisors, take steps to appropriately introduce student teachers to these difficulties and complexities so that valuable learning takes place. Another critique of our two ideas is that mentor teachers are ultimately responsible for students learning mathematics in their classroom. When mentor teachers take on the added responsibility of hosting a student teacher, they must make decisions that are best for their students. In some cases these decisions may limit the extent a student teacher can be treated as a colleague. For example, there may be instances where a K-12 student’s personal or medical history may prevent the mentor teacher from sharing the reasons for making particular decisions. We recognize these situations and suggest that mentor teachers use their professional judgment to manage them. Three Questions Universities and K-12 schools share the goal of providing meaningful learning experiences for prospective teachers during student teaching. Yet, a

Ginger A. Rhodes, Jeanette Phillips, Janet Tomlinson & Martha Reems

closer look reveals possible differences in what one means by a “meaningful learning experience.” These differences tie into one’s beliefs about the purpose(s) of student teaching. This leads to our first question: What is the purpose of student teaching? The answer to this question might be different for student teachers, mentor teachers, and university supervisors. Possible purposes include being enculturated into schools, learning to manage students, practice ideas learned from university courses, learn how students think mathematically, and to experience all aspects of teaching. In some instances, differences in purpose for student teaching create a divide between mentor teachers and the university. For example, some mentor teachers may be concerned with prospective teachers being successful in the present moment, whereas universities maybe concerned with prospective teachers being successful over their careers. We believe a balance between preparing prospective teachers for the moment and for the future is necessary. Finding the balance is a negotiation that can only happen when the vested parties consider and communicate their beliefs about the purpose of student teaching. Through these negotiations, a common understanding for the purpose of student teaching can be developed. We are not suggesting that the vested parties agree on one common purpose, but they should be aware and understand each other’s purposes. A university teacher preparation program is the initial training experience aimed at preparing teachers. Many of these preparation programs expect student teachers to teach in a manner that differs from their prior conceptions of teaching. Likewise, mentor teachers are asked to participate in a student teaching experience that differs from their own experiences. For example, mentor teachers are asked to have professional conversations with student teachers in order to make instructional decisions explicit. In many instances, these conversations are new for mentor teachers. Mentor teachers are also asked to co-plan and co-teach lessons. The idea of co-planning and coteaching can be interpreted and used in several ways. For example, in some co-teaching situations one teacher may be responsible for leading some portions of the lesson while the second teacher leads others. In another version of co-teaching, one teacher maintains the lead throughout the lesson while the second teacher is a helper. The helper may pass out materials and work with smaller groups of students. In a third way of implementing co-teaching, both teachers can lead by taking equal roles throughout the lesson. A way that might help one think about both teachers leading is to 45

consider the teachers jointly having a conversation with students. Sharing in professional conversations, co-planning, and co-teaching are new ideas for many teachers; therefore many teachers have limited views of how to implement these ideas with student teachers. Situations like this lead to our second question: How can the university support mentor teachers in their efforts to develop their own mentoring skills? Schools are faced with the challenge of providing a high-quality mathematics teacher in every classroom. The shortage of such teachers is evident, as there are many classrooms where teachers are either not certified or have substitute status. Similarly, universities face the challenge of placing student teachers in positive learning environments supported by high-quality teachers. Though student teachers can learn from both good and bad experiences, the learning is different. Learning what not to do in a classroom is different from learning what to do. The assumption that good university preparation trumps a poor student teaching experience is unfounded, as Frykholm (1996) found mentor teachers have a greater impact on instructional practices of preservice teachers. This observation lays the foundation for our final question for teacher educators: What are some options if a student teacher is placed in an environment that is not ideal for their learning? Put another way, what options are available to teacher educators if a mentor teacher immediately gives full classroom responsibility to a student teacher without providing any guidance? We can all agree that this is not the kind of environment that we want for our student teachers. Is the best option to remove the student teacher? If so, where does the student teacher go? Are there other ways to manage the situation? Conclusion At first glance, the ideas we highlight in this paper do not seem difficult to implement, but there are many instances where student teachers do not feel comfortable trying new ideas or do not regularly participate in professional conversations. We feel these student teachers are disadvantaged.

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When we consider the student teaching experience, we recognize the larger purpose is to prepare prospective teachers for their future work as teachers. Many times, however, mentor teachers voice their excitement for hosting a student teacher for the reason that they are interested in learning new teaching methods and technologies. We want to draw attention to the professional growth opportunities for mentor teachers and university supervisors during the student teaching experience. Mentor teachers and university supervisors can gain much more than new teaching methods and technologies. For example, when mentor teachers make explicit their instructional decisions, we believe this encourages them to thoughtfully think through and possibly reconsider those decisions. Ultimately, student teaching should be viewed as a learning opportunity for everyone involved. In our final comments, we wish to emphasize the importance of developing strong relationships between the universities and K-12 schools. The answers to the questions posed above should come from both institutions, as both institutions make valuable contributions to the student teaching experience. It is our responsibility as educators to develop these relationships so that we may provide each prospective teacher with a worthwhile initial experience as a mathematics teacher. References Frykholm, J. A. (1996). Pre-service teachers in mathematics: Struggling with the Standards. Teaching and Teacher Education, 12, 665â€“681.

1

This work was made possible in part by Grant ESI0227586 from the National Science Foundation to the Center for Proficiency in Teaching Mathematics at the University of Georgia and the University of Michigan. Any opinions, findings and conclusions, or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation (NSF).

Mentor Teachers

CONFERENCES 2007… MAA-AMS

New Orleans, LA

January 5-8, 2007

Irvine, CA

January 25-27, 2007

San Diego, CA

February 22-25, 2007

Cleveland, OH

March 1-3, 2007

Atlanta, GA

March 19-21, 2007

Atlanta, GA

March 21-24, 2007

Chicago, IL

April 7-11, 2007

Keele University, UK

April 11-14, 2007

Fredericton, Canada

June 8-12, 2007

Penang, Malaysia

June 18-22, 2007

ICTMT8 8th International Conference on Technology in Mathematics Teaching

Hradec Králové, Czech Republic

July 1-4, 2007

PME-31 International Group for the Psychology of Mathematics Education

Seoul, South Korea

July 8-13, 2007

Salt Lake City, UT

July 29-August 2, 2007

Warsaw, Poland

July 31-August 3, 2007

Joint Meeting of the Mathematical Association of America and the American Mathematical Society http://www.ams.org AMTE Association of Mathematics Teacher Educators http://amte.net SIGMAA on RUME Special Interest Group of the Mathematical Association of American on Research in Undergraduate Mathematics Education http://cresmet.asu.edu/crume2007/ RCML Research Council on Mathematics Learning http://www.unlv.edu/RCML/conference2007.html NCSM National Council of Supervisors of Mathematics http://www.ncsonline.org/ NCTM National Council of Teachers of Mathematics http://www.nctm.org AERA American Education Research Association http://www.aera.net Mα

The Mathematical Association http://www.m-a.org.uk/resources/conferences/index.html CMESG Canadian Mathematics Education Study Group http://cmesg.math.ca ICMI – EARCOME4 The Fourth East Asia Regional Conference on Mathematics Education http://www.usm.my/education/earcome4

http://www.igpme.org JSM of the ASA Joint Statistical Meetings of the American Statistical Association http://www.amstat.org/meetings/jsm/2007/ First Joint International Meeting with the Polish Mathematical Society http://www.ams.org

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In this Issue, Guest Editorial… Mathematics for Teaching or Mathematics for Teachers? LEE PENG YEE Uncovering Algebra: Sense Making and Property Noticing DAVID SLAVIT Preparing Elementary Preservice Teachers to Use Mathematics Curriculum Materials ALISON M. CASTRO Direct and Indirect Effects of Socioeconomic Status and Previous Mathematics Achievement on High School Advanced Mathematics Course Taking MEHMET A. OZTURK & KUSUM SINGH Efficacy of College Lecturer and Student Peer Collaborative Assessment of In-Service Mathematics Student Teachers’ Teaching Practice Instruction LOVEMORE J. NYAUMWE & DAVID K. MTETWA In Focus… Mentor Teachers’ Perspectives on Student Teaching GINGER A. RHODES, JEANETTE PHILLIPS, JANET TOMLINSON & MARTHA REEMS